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dtheriault-- I don't know exactly what "could gravity move outward" mean, but no... Gravity is only an attractive force. The fact that the mantle is more dense than the core could create some interesting scenarios for the planet, but any observer on or near the planet would still feel a pull towards the planet. It doesn't matter where the mass is located in the planet as long as it is there, it will pull "downwards". I don't think anyone can say why exactly it only pulls and doesn't push; there may be some theories on the edge of physics that may attempt to explain this, but I imagine it is still pretty unknown.
Charlatan-- gravity is the attraction of everything around us with mass to everything around us (and us) with mass. Sorry to get all tautological, but that is simply what "gravity" is defined to be. Gravity is one of the forces that act on bodies in the universe... To go any further (at least with what I perceive to be the intention of your question) would probably require either to go into metaphysics or more complicated theories like the gauge theory that I don't really understand anyway and probably wouldn't really get to the root of your question.
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