What is the largest sphere of liquid water that could exist in space held together under it's own gravity? What is the largest theoretical sphere of liquid water that could exist in space held together under it's own gravity? I've always wondered if a planet the size of earth could exist as a single volume of water? If in orbit of a star (which I guess it would need to be to remain a liquid), would the star's UV radiation evaporate such a watery world? Do the laws of physics permit enormous volumes of liquid water to float around in space?
 A: A planet all in liquid water is very unlikely. But all water is still physically possible. You just got to have enough mass of water to hold the whole body together, by gravity. 
Different physical states of water will exist however, the further you are away from the center of gravity will be in a gaseous state (water vapor), reducing in density, and pressure as we go further towards the further points in the outer atmosphere and outer space. 
The further you go towards the center of gravity however, pressure, along with temperature will increase until it gets high enough to very likely form super critical water. That's a state where water will be in a state where distinct liquid and gas phases, doesn't exist.
On the other hand a planet in all liquid state is not possible since gravity will not allow pressure to be equal in all parts of the planet, resulting in this distinction of the phases.
Also if the planet had a weak enough gravity to allow the vapor to escape gradually, the planet will gradually evaporate from the outside until completely depleted. The water vapor surrounding the liquid body I described earlier acts as a buffer zone between outer space, which has no pressure, and the denser areas below.
The UV will give energy to the water vapor molecules and will tend to escape, but you can overcome such effect by a strong enough gravity. Since hydrogen is the lightest of gases, the earth's gravity can't hold hydrogen molecules, thus you can't find those in our atmosphere, contrary to other gases who needs more energy to escape.
A good example of a planet in a state similar to your description here is Uranus. The planet consists mostly of liquid and gas, although not entirely of one kind of them.
Such planet also is not likely to naturally consist of all pure water, as celestial bodies like meteors and dust, etc.. Will fill it with impurities, thus breaking the rule you just pointed out. 
In order for a star to be formed you will need some 70+ times the mass of a gas giant like jupiter to form. So as long as you are well below that kind of mass you're fine with your planet.
A: Planets made mostly of water almost certainly exist, and at least one may have been detected. 
However, such bodies will not be made entirely of liquid. In fact they will be mostly solid, even if the temperature is very high. This is because water can form more than one form of ice. The kind we're familiar with forms at low temperatures and is less dense than water. But there are several other kinds that form at very high pressures instead. These "exotic" ices are heavier than liquid water, so will sink to the bottom. You can get a feel for the pressures involved by looking at a phase diagram of water, e.g. here or here.
Earth's oceans aren't deep enough for such pressures to exist, but if the whole planet were made of water, the planet would be composed mostly of "exotic" ices, covered by a relatively thin layer of liquid (though this would still be a lot deeper than Earth's oceans). In fact, Europa, Titan and many of the moons of the outer Solar system are probably more or less like this, except that they have a layer of normal water ice on top of the liquid ocean, because they're so cold.
Another answer mentions the possibility that there would be a smooth transition from water vapour to liquid water, rather than a clear ocean surface as there is on Earth. Whether this is true or not depends on the temperature. The critical point for water is at about $647\,\mathrm{K}$, or $374^\circ\mathrm{C}$. If the temperature is higher than this then there will not be a phase transition between the atmosphere and the liquid, but if it's lower then there will, just as there is on Earth.
With those details out of the way we can address the question of stability. Your original question was what's the largest body of liquid water that can hold itself together under its own gravity. However, I don't think such a body can exist at all. If it did, it would have to be rather small in order to prevent the formation of exotic ices in its interior. But that would mean its gravity would be quite low, so it wouldn't have an atmosphere. The low pressure at the surface would cause the water to evaporate or boil, and the molecules of the resulting gas would easily be moving fast enough to escape the body's gravitational pull. (Of course, things are different if we allow the surface to be frozen - see the outer Solar system moons - but I'm assuming you're specifically interested in a body with a liquid surface.)
However, if we allow the interior to be composed of exotic ices rather than liquid, it gets a lot easier. Then all we need is for it to be big enough that it's escape velocity is higher than the velocity of water vapour molecules. Anything from around Earth-sized upwards should do the trick. I don't think there's an upper limit to the size, except that if it's really, really big then nuclei in its core will start to fuse and it will turn into a star.
You mention UV radiation causing a loss of the water, and this is an issue for a body the size of Earth. UV radiation can split water into hydrogen and oxygen, and the hydrogen atoms have a much higher velocity, making it easier for them to escape into space. On Earth we're fortunate to have an oxidising atmosphere, which tends to turn the hydrogen atoms back into molecules before they can escape. If it wasn't for that, the Earth's oceans would already have disappeared.
However, if we start with a pure $\mathrm{H_2O}$ planet that's big enough, eventually it too will have an oxygen atmosphere. Not because of photosynthesis, but simply because as the hydrogen escapes it leaves oxygen behind. Given sufficient time I would expect this to lead to a protective oxygen atmosphere that prevents further escape of hydrogen. 
The only detail left is how to get a surface temperature that's in the right range for the surface to be liquid. This depends on the distance from the star, but also on the composition of the atmosphere. On Earth, the atmosphere's composition is mostly due to the action of the biosphere, which keeps the temperature regulated in just the right range for water to be liquid. Perhaps it's possible to imagine life on such a water world, in the form of photosynthesising algae-like organisms, which might play a similar role.
