Wouldn't a droplet of liquid (say water) floating around in space become spherical and "in hydrostatic equilibrium" due to surface tension but not gravity?
Strictly speaking (as far as I know), hydrostatic equilibrium applies whenever a fluid balances external body forces with the pressure gradient. From Wikipedia:
In continuum mechanics, a fluid is said to be in hydrostatic equilibrium or hydrostatic balance when it is at rest, or when the flow velocity at each point is constant over time. This occurs when external forces such as gravity are balanced by a pressure gradient force.
I think the concept happens to be most frequently used in areas where gravity is the external force, but it could in principle be anything else. So, though I stand to be corrected, I think a droplet isolated in space long enough could be said to be in hydrostatic equilibrium, even though the most relevant force is the surface tension, rather than gravity.
Surface tension tends to draw the water into a nice sphere.
Yes, a droplet of water would do that.
However, at the planetary scale, the contribution of superficial tension is negligible. At that scale, gravity dominates all other forces.
In fact, you could model a planet as a loose mass or rubble, kept together by gravity alone, and the model would be very close to reality.
We have found instances of asteroids that are mostly rubble piles, but spinning fast enough that they would fly apart if bound only by gravity. The most likely binding force is static charge, although some sort of glue is not out of bounds. I don't know if either of these would be called "hydrostatic", but if the question is really "Is gravity the only force that can bind a non-solid object together?" the answer is no.