could there be a blob of liquid water in space the size of, say, a planet?
It's pretty unlikely, but yes, theoretically it's possible.
Could the water being ejected by that black hole ever condense into something of the size I'm describing?
That would be one of the very few scenarios where something like this could form. Maybe. It depends on the density fluctuations of the water ejecta. You need a condensation center to gather a large mass of water and get the planet started.
Would it boil over immediately?
A cloud of stuff contracting under its own gravity will definitely heat up. But here's the thing - it does not matter whether the water is solid, liquid or gas. It would collapse just the same. Gravity is stronger. Doesn't matter whether is cold as ice, or boiling like crazy.
Once it's a small sphere of water in some form, more complex phenomena will take over. See below.
Would the elements near the middle heat up and maybe fuse?
Maybe some heat will be provided by the initial collapse. Maybe it does have some radioactive impurities which will heat it up. But anyway, over a very long time it would tend to cool down.
In general, the core would be an exotic form of high-pressure ice, no matter what the temperature - water is solid at very high pressure, even if you elevate the temp a lot. Above that there will be a layer of liquid water, if the whole planet has enough warmth, or just plain old ice if it's too cold altogether. The surface might be solid again, cold ice (if the planet is wandering alone in space), or liquid (if it's close to a star). There may be a water atmosphere above, either wispy and thin (cold planet) or thick (warm planet).
Roughly how large a body of water could you have that's stable given that it could be protected from the vacuum?
When the escape velocity of water molecules is bigger than the average thermal speed, the body is stable. In other words: Warm planet - needs to be bigger to keep the water in. Cold planet - it can be smaller.
A small chunk of ice could survive for quite some time in outer space before sublimating to nothing. A Moon-size blob of ice would probably be stable forever. But if there's warm water on the surface, and a thick water atmosphere, it would probably require an Earth-like mass (and gravity) to keep the water from escaping.
And just for fun, supposing the body of water was "bootstrapped" with primitive life or the ingredients for it, could it potentially support life?
If it's life that doesn't need dry land and ocean bottom to survive, then yes.
There's a type of exoplanet that is similar to what you describe. Not identical, but kind of close: