Gravity's Role in Equilibrium
Gravity pulls everything toward the center of the Earth. It creates the weight of the water column, which generates pressure. However, gravity also holds the Earth's crust together. The crust is a solid, continuous layer that supports the weight of the ocean. There is no 'hole' in the center of the Earth for the water to fall into. Instead, the water rests on top of this solid support, much like a book resting on a table. The table supports the book; the crust supports the ocean.
No Central Void
A common misconception is imagining the Earth as a hollow sphere with a drain at the bottom. In reality, the Earth is solid (with a liquid outer core, but that is separated by thousands of kilometers of mantle). The ocean basin is a depression on the surface, held up by the strength of the lithosphere. Gravity pulls the water down, but the rock pushes back up with equal force.