Why doesn't the earth's surface collapse onto itself?
For the same (an more prosaic) reason milk skin floats over boiling milk - because it is lighter than the layer below. During the formation of the planet it went through a process called Iron Catastrophe, were a large parcel of the denser materials like iron and nickel sunk to the core, leaving the lighter materials close to the surface.

Nice graph with relative densities follow:



Fission contributes to keep our internal engine super-hot and well-oiled, allowing for our metallic core to spin and generate our magnetosphere. Eventually we'll run out of heat, the outer core will solidify and stop spinning.