What stops gravity from pulling water to the center of the Earth?

The same thing that prevents gravity from pulling you to the center of the Earth; electromagnetic repulsion, or more properly, electron degeneracy pressure.

Before describing electron degeneracy pressure, I want to point out that a lot of water probably does make it at least as far as the mantle, particularly in the form of subducted oceanic crust. It's true that water can saturate the crust almost completely as long as pressure, temperature and porosity allow it, but by the time the water hits the mantle its further subduction is strongly resisted by the pressure and temperature of the liquid rock in the mantle. Without actually drilling, it's pretty difficult to discern whether or not there is actually any water in the core, but it would be a safe assumption to say that it's cycled between the upper mantle, crust and atmosphere.

Electron degeneracy pressure is the result of electromagnetic forces acting between atoms. Atoms are surrounded by their electrons, which creates a field of negative electric charge, repelling other negative charges. When two fields come into contact, they can be forced to interact through extremely high pressures and temperatures, but under normal conditions it is not possible for electrons to occupy the same spatial area, and this resistance greatly outweighs the forces with which they are typically pushed together. So, when you or water are sitting on the surface of the Earth, the Earth's electrons are effectively pushing back against you to keep you where you are. Likewise, once water has reached the saturation point in a given area, any further water will "pile up" due to repulsion from the molecules sitting below them, including other water molecules.