Bsrgvty

You won't be able to have this in a universe that uses our physics.

The degree to which space itself expands or contracts is related to the cosmological constant and the density matter within that space.

A universe full of water is going to have an insane matter density. Which can be balanced by a large cosmological constant.

The problem arises in that this balance is not stable -- at an iota greater, the higher cosmological constant causes runaway expansion, and an iota less it causes runaway collapse.

In our universe, this runaway expansion is going on; the universe is effectively in a "gas phase", and decreasing the density just causes galaxies to move apart.

In water, density would lead to the fluid boiling. The resulting pockets of vapour would float towards each other (as being lower density), and the remaining fluid would fall together, and everything would be extremely far from static. The clumps of water would collapse inward radiating insane amounts of energy and form black holes.

Now, the universe we are in was once as dense as your water universe; in this period, it had an insane inflation rate. That inflation rate was fast enough that there wasn't time for things to become non-uniform. The universe was also insanely hot (hot enough that helium was too hot to stay helium!). It cooled rapidly, and at around 11kK it got cool enough to allow for He to be stable, and then it continued to cool.

The ridiculously tiny ripples in density it had while dense was enough to form huge voids and galactic clusters as the density kept falling.

So no, at no time can you have an extended "water universe" using our physics.

It turns out I could be wrong. "Unstable" doesn't mean immediately unstable.

A pressure in the gigapascal range can have liquid water. Given our current universe expansion, that pressure would take billions of years to relax.

Unfortunately that adiabatic expansion (there is nowhere for the heat to go, other than other places with heat) will be doing work. Naive Newtonian physics makes the heat change pretty large. I don't know, cosmologically, if the pressure release/work would do strange things to the cosmological constant/expansion rate.

This still leaves another problem; the fluid is not going to be uniform; our universe doesn't permit uniformity. You'll get areas of different density, and those will attract and "repel", causing increased density differences.

In a gas-phase universe collapse is very enthropically encouraged; the falling gas heats up, radiates, and repeats until held back by something as strong as nuclear fusion or a phase change to solid or liquid or electron degenerate or neutron degenerate or becomes a black hole.

A liquid phase universe is going to behave quite differently during gravitational collapse due to the incompressibility of liquid. Tiny decreases in size (and increases in density) will absorb huge amounts of energy.

Equations of state

The "ocean" should be described by an equation of state that relates thermodynamic variables (e.g. density $rho$, temperature $T$, pressure $p$, etc.) to one another. Different equations of state are valid in different regimes; some are valid at high temperatures and pressure, while others are valid at low ones. I think it's not particularly worth going over them here, but the usual (approximate) one for liquid at reasonable temperatures and pressures involves just density:
$$rho=textconstant$$
that is, liquid water is essentially incompressible. By looking at Bernoulli's equation (which works for incompressible fluids), we can see that in the case of a static fluid in a vanishing gravitational field, the ratio of pressure to density is constant, and therefore the pressure must be constant, too. Note that when I say "constant", I mean that a quantity does not change much if you change other, related, thermodynamic quantities. Change the pressure of water slightly and you won't significantly change its density.

If we were to increase the ambient pressure inside the fluid, the assumption of incompressibility would be invalid and we would have to pick a different equation of state.

Pressure and pressure gradients

I think there are two quantities that may have been mixed up here: fluid pressure and the pressure gradient inside the fluid. I'm not aware of any realistic equation of state - certainly not one for liquid water - that involves vanishing pressure for a fluid with non-zero density. Pressure typically arises from interparticle interactions, and as the density is decidedly non-zero, there's no reason for that pressure to go away.

Now, what will go away is the pressure gradient $vecnablap$. In a fluid in hydrostatic equilibrium experiencing a gravitational acceleration $vecg$, the pressure gradient is
$$vecnablap=rhovecg$$
and in our case, as $vecg=0$, so is the pressure gradient.

A solid object

Now, say we have a cubical box of side length $L$ embedded in this fluid. All sides of the box will experience a force of magnitude $F=pL^2$, where $p$ is pressure; the force on each side is pressure times the area of a side, which follows from thinking about pressure as force per unit area. However, because there's no pressure gradient, these forces oppose each other equally, and so there will be no net force on the box.

The box may be deformed by the pressure exerted on it on all sides, but it will not experience a net force in any direction.

Cosmology

Several users (Yakk, Acccumulation, and Joe Bloggs) have noted that this universe will likely not remain uniform, but will suffer one of two gravitational instabilities:

• It may collapse in on itself in a Big Crunch if the density is larger than the critical density, which is actually quite low. If the critical density is the same as in our universe, then this universe is far too dense, and will almost certainly collapse.


• 
  Small perturbations, like the quantum fluctuations seen in the early universe, will eventually cause structure formation in much the same way that our own universe first formed inhomogeneities. The characteristic scale of these fluctuations is
  $$lambda_J=c_ssqrtfracpiGrho$$
  where $G$ is the gravitational constant and $c_s$ is the speed of sound. In this universe, $c_sapprox1481text m/s$, and so $lambdaapprox10^7text m$, which seems rather large; the characteristic growth time will be $tauapprox10^3text s$.

A similar question might be: in an infinite universe filled with hydrogen, what gives rise to pressure?

If this seems an odd comparison to draw: consider that , near the beginning, our universe was pretty much homogeneous hydrogen. It was also very small, but that doesn’t matter for this answer.

The answer I think you can pull out here is that even if your universe starts off ‘homogeneously mixed’ it will not remain so. Your universe is only homogeneous on a large scale. Seen up close there will be small-scale variations.

These variations will, over time, lead to larger variations. A small clump of hydrogen will, ever so slightly, pull in more hydrogen. The once uniform gas will start to pull itself into filaments as parts of it gather into more massy areas. Those filaments will split into beads, drawing the rest of the hydrogen out of the less massy areas (which will come to resemble vacuum quite quickly). Those beads will gather in size and mass and gravitational potential until they auto ignite in a blaze of nuclear fury. From there it’s only a matter of time until you have planets, atmospheres and a much more traditional definition of pressure.

Your universe starts off much more oxygen rich, so all the stars will start off further along the main sequence (if indeed they’re on it to begin with). Other than that you can expect the same things to happen on cosmological timescales. Somewhere a small chunk of water will gather (pretty much randomly) at a higher density than the water around it, disturbing your uniform gravitational field. From there it’s only a matter of time until everything gets pulled into strings of clusters of galaxies of stars.

It might take a while though...

My logic is the same as yours; in an infinite universe entirely filled with liquid of a uniform density there's no gravitational variation. The universe is effectively just filled by a thicker form of vacuum.

Once you introduce a gravitational anomaly, in the form of an object of higher density, it gets very different very quickly. Since the "vacuum" of this universe actually has considerable mass per unit volume a high density object will cause the universe to begin accumulating at that gravitational centre turning into a giant ball of water that eventually (in the space of a few seconds) collapse in on itself under it's own mass creating a giant blackhole.

A lower density object won't really do anything but sit there; unless there are ambient currents in which case it will get carried in them like any other particle.

An 'Infinite Ocean' eventually transforms into a universe like our own (planets, stars, black holes) : because of the random motion of particles

A uniformly distribute universe of water may start with the stated uniform distribution, but it won't stay that way for long. There is a simple reason for this, water particles are not stationary.

Imagine you start with a uniformly distributed infinite ocean. Water molecules will move around "randomly" because they are energetic, and energetic particles move around. This means they will eventually move into a position where the distribution of the water's mass is not uniform.

At this point uneven gravity will begin to act on the water column. Eventually the water column will be completely uneven and we will end up with a very perceptible directional gravity in different pockets of the universe. This all culminates in localized blackholes forming, as an enormous mass of water collapses in on these local points of high gravity. I can't say how long this would take, but for sure it would happen eventually.

However not all areas of high gravity will have enough mass to form blackholes, so in some areas there will be planets made entirely out of water, and water based stars (instead of hydrogen based ones). These stars will then make metals, and other elements using fusion. All these bodies are probably rogue, or else orbiting around blackholes.

Even if we try to force this universe to work as a lasting infinite ocean. For example by assuming all water molecules have the exact same orientation, energy, and therefore travel parallel to each other. Quantum effects will still cause the behavior of these water particles to change over time so that they are not all the same. Once this happens, its then a slippery slope back into the black hole(s) filled universe that we just discussed.

In a vein similar to Yakk's answer, the issue you are concerned with is pretty much the same as the issue of density in our universe, except many orders of magnitude greater. Our universe also is filled with fluid (mostly hydrogen), but at many, many orders of magnitude lower density. We can then ask: for our actual universe, do we have equilibrium, since every point has an infinite amount of hydrogen pulling it in every direction? Or should the universe collapse, since there's a contracting force at every point? The overwhelming consensus is that absent a cosmological constant, it would be the latter; there would be no static universe. A universe can be expanding, and having its rate of expansion slowed by the contracting force. It can be contracting, and having its contraction accelerated by the force. But (again, absent a cosmological constant), it can't be stationary.

Seeing as how any cosmological constant in our universe would be of roughly the same order of magnitude as the density of matter, and your universe would have vastly higher density, we can conclude that either your universe has a vastly stronger cosmological constant, or it has extremely massive pressure. I would expect the pressure to be so large that the matter would quickly cease to be water.

I think it's important to note that your assumption that infinite, uniform mass density implies zero gravity is not necessarily a valid one. The second answer of this physics stack exchange article goes into more detail, but the general idea is that infinite universes often mess with the math that goes into calculating gravity, since our usual solutions to such equations involve "nice" behaviors at infinity (ie, potential is asymptotically zero in Newtonian gravity, or the metric is asymptotically minkowskian in GR). When these conditions aren't met, you tend to run into the problem that your equations no longer have unique solutions. This is a problem because the whole goal of science is to come up with models that predict what will happen if you perform some experiment-- it's not very useful if your model has an infinite number of different predictions.

TLDR: the scientific models currently in use can't really give a meaningful answer to this question.

You could just claim it’s a universe with no Higgs boson or Higgs field and so no mass and thus no effect of gravity, theoretically? But that would have other issues. Depends how fully baked the answer needs to be.

I just am trying to say you could alter a single or maybe a couple elements of physics in this universe to claim something for your water universe and really not many could dispute it as we don’t really understand most of the theoretical physics around it.

We’ve proven the existence of the Higgs particle, but we don’t really understand it, the Higgs field, or even gravity really... so I’d say just change the physics of that universe to make it what your proposed idea needs.

Just a thought.

First, I'll note that the water doesn't need to be infinite.

The gravitational force that causes pressure can propagate through water only at the speed of light.

So assuming you started out with a pressure gradient of 0 throughout the water blob, and the age of the water-blob in seconds is less than the shortest distance to the edge of the blob in light-seconds, then the pressure where you are measuring will be unaffected by the fact that the water is not infinite.

Assuming it's in energetic equilibrium at 4°C, or at least that it is not in gaseous form, and you plop a lump of something dense in, the dense thing shouldn't form a black hole. It should just form a pressure gradient where the pressure at each point cancels out the force of gravity from the object, plus the gravity from the slightly-increased density. Over time, as the existence of this dense object propagates away at the speed of light, everything in the universe arguably moves a little closer to it, but not detectably for most of the universe as the effect dies off with the inverse cube of the distance.

This means an appearing object causes an outgoing wave of negative pressure, as stuff is sucked towards it a little.

I think this also means that not only no black holes could form, but no planetary systems could form - no stars, planets. No accretion of any objects through gravitation at all, since the repellent effect of the pressure differential should greatly outweigh the effect of gravity (citation admittedly needed).

Note that this should hold true for universes filled with solids, as well as with liquids.

I say no pressure in an infinite ocean.

If the ocean was truly infinite, there would be no center of mass.

Thus, no gravitational attraction would pull the water to any one point to cause the pressure.

The only thing that would cause pressure is how much water there is compared to the size of the universe. If space is infinite and the amount of water is infinite, it then depends on which infinity is bigger.

If Infinity(water) is greater than Infinity(space) then pressure would increase, the ratio indicating how much pressure is exerted on any non-water forms in that universe. If Infinity(water) is less than Infinity(space), the water will be water vapor instead of water. For certain ranges of I(w) < I(s), water may exist in some transitional state between liquid and gas.

Okay, I tried resisting posting... but I'm going to give a shot at answering because I didn't agree with a lot of the answers. So, instead, I'm going to answer by proposing thought experiments.

T.E. #1: Vessel of water in space

Imaging you're aboard the ISS. You've got a cylinder filled with water that is plugged on one end. The other end has a plunger that's loosely containing the water.

What happens when you press down on the plunger?

Well, the pressure in the water builds - you're putting pressure on it. But, generally, the water is staying the same volume - because water is pretty darned uncompressable. Water at the bottom of the ocean only manages to compress water 5%.

And if you press really really really super-humanly hard? Then the water actually condenses to a solid. Which really doesn't compress well.

T.E. #2: Filling ISS with Water

Forget the vessel. Let's blow out all the air, patch up the hull, and replace the vacuum with water.

What's the pressure inside?

Well, when you first start filling it with water, the water will boil into steam - if you look up a tri-state chart of water, you'll see that for low enough pressures, you simply can't have liquid water: it'll either boil or freeze.

But as more water is added, something weird starts to happen. Eventually, the pressure of steam will get high enough that water starts re-condensing and the water you're adding won't be boiling off. If we were in an environment with gravity, we'd have a layer of water on the ground and a layer of high-pressure steam above it. But gravity isn't in this picture (ISS!) Instead, we have water and steam intermingling - and since water has some good surface tension, we'll generally see globules of water meandering through the steam.

As we add more and more water, the pressure goes up, the volume taken by steam goes down, until eventually the entire ISS is filled with water with no steam. And that water is just above the condensation pressure for water. In real-world terms? It'd be what water is like at the surface of an ocean/pool/puddle.

... and as you add more water? The pressure starts going up. You're adding more water molecules, but not giving them any more space. And given how incompressable water is, the pressure starts to climb pretty rapidly.

And like the plunger situation (and assuming ISS was built out of infinitely strong material) you'll have a situation where, when you added enough water, you'd cause the water to start solidifying due to presure.

T.E. #3 - Giant Plunger

Imagine you've got the same cylinder and plunger from the first thought experiment, but this time: They're huge. Once again, you press down superhumanly hard, and barely condense the water into ice.

What happens when you let go of the plunger?

On one hand, Ice VI and Ice VII have a density either 30% or 50% greater than regular water. So if a chunk of regular water would be pulling with 10 newtons of force due to gravity at N meters, a chunk of Ice VI or Ice VII would be pulling with 13-15 newtons.

Now, think how little this difference of gravity would be... versus how great the pressure difference would be! Imagine how much force a column of water 100 kilometers deep would push down on you? That's how much force the ice would be trying to exert in order to expand.

In other words, the plunger would go flying away from the ice - the ice would be pushing with immense force - far more than its slightly higher gravity would hope to counteract.

Putting Everything Together

As the OP guessed at: long distance gravity wouldn't matter much. If the universe was simply infinite ocean, the gravity would have no net direction/force.

But that's not to say there wouldn't be pressure. Like our ISS in Thought Experiment #2, it all depends on how many water molecules there are in a given amount of space. Not enough: no ocean (its all steam.) Too much: sea of exotic ice. Just barely enough: water with pressure like you'd find at the top of the ocean. Almost too much: water with pressures greater than any ocean trench.

Now, something stirs up 'space' - such as the pull of a local dense object. You've effectively got a plunger pushing downward on your water. If that object isn't very big? You're just lightly pushing down on the plunger - until the pressure from the water is equal to your force. If the object is really dense and massive? Then the plunger is pushing down hard enough to solidify the water around the surface to exotic ice... but ultimately it's like our T.E. #3 - that ice has enormous amounts of force that it's using to try to expand.

So, TLDR: The water's going to be pretty uniform: water pressure is much stronger than increased gravity from denser water. Even dense non-water objects will probably simply have a gradient of increasing water pressure - and possibly exotic ice - around them.

Finally, to the OP's question: what would a solid experience in that environment? Figure out what the pressure would be (dependent entirely on how much matter there is per volume) and then ask yourself: how would an object under the surface of the ocean act (assuming the object has the same density of water, since gravity doesn't play a role.)

Since this is science-based, consider your frame of reference. You can have a universe of water by using some tricks, like letting the reader make some assumptions. Thus:

"The Universe" to us is quite big. But to a jellyfish swimming in your water-universe then its small to us, but still quite big for the jellyfish. To an amoeba the inside of the Jellyfish is an entire universe.

To get what you want in your story, then leave the reveal toward the end, that the scale is smaller-than-human scale. The water universe could be a water planet with a relatively small diameter, but characters who are sub-millimetre tall. They might speculate about "what lies beyond the known universe" just as humans do.

This will provide a framework for "why the water pressure is not crushing"

Every point would be the center of the infinite universe and therefore experience no gravitational force. One equilibrium in which this universe could exist would be one in which all the molecules are in cubic symmetry (each one equidistant from all its nearest neighbors). So such a universe would essentially be one gigantic crystal.

Putting an object in it would disrupt the mass symmetry and make that object the center of mass of a gargantuan body of water and eventually lead to stars and other stuff.