[apache]

"Hmmm. I'd say that if the net gravitational force on an object is zero, that by definition, all the gravitational forces have canceled out. Gravity is not like pressure in the example you give. For a small object (relative to the size of the planet), say a person, if the net gravity is zero for that person, every point (ie: every cell, every molecule) on/in the person will have a net gravity of zero. The tidal differences between, say, points on the person's left and right arms would be too small to matter."

Here is something I can finally dispute with certainty. Using Newton's second law (assuming it applies in the center of a planet/other things), we know that the sum of all the forces acting on the object equal its the change in momentum of the object. Ok, so adding up everything via integration methods, we find that the net force on the object is equal to 0. From this we figure that the change of momentum is 0. So if the object is at rest, it will stay at rest, and if it is moving, it will continue to move (Newton's first law). Thus, we say the forces acting on the object are in equilibrium.
However, equilibrium does not mean the forces 'cancel out'. Now that we found out the forces acting on the entire object, we can then look at internal forces. So, lets cut the object in half (not actually cut it, just an imaginary cut to examine the forces on the inside). Ok, now we know that the entire body was in equilibrium, so this section of the object must also be in equilibrium, otherwise the two halves of the body would move away from eachother. However, by cutting it in half, we have eliminated the gravitational forces acting on the other half of the object (they are still there, but since we are analyzing the other half, we only get half of the gravitational forces. So what we see is that for this object to be staying in one piece is that there must be internal forces to counteract the gravitational forces (Newton's third law). This means there must be internal stresses.
From this point, knowing the internal stresses, you can compare them to the known strength of the material the object is composed of, and if the stress is too high, the object is ripped apart, if not, then it stays together.
This is much like a tug of war. Just because the rope may not be moving if each side pulls with the exact same force, that does not mean that the rope does not experience force. If those forces pulling the rope are too great for the strength of the rope, it will rip apart.