Why can gravitational forces be based on the center of mass. Due to the fact that gravity is related to the square of the distance should not the gravitational sum of every particle exceed the force when calculated by the center of mass.
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Well that depends. As always, you should be very careful with such reasoning as
Due to the fact that gravity is related to the square of the distance should not the gravitational sum of every particle exceed the force when calculated by the center of mass.
because this is a problematic statement. In general your (Newtonian) gravitational potential is given by the Poisson equation: $$\vec\nabla^2\phi=4\pi G\rho(\vec x)$$ Which can be solved via the green function for the Laplacian: $$ \phi=\int{\frac{G\rho(\vec x')}{|\vec x - \vec x'|}\mathrm{d}^3x'}+\mathrm{C} $$
and this potential is related to the force of gravity by: $$\vec a=-\vec\nabla\phi$$ You cannot say in general that this results in an inverse square law. In case of a spherically symmetric mass distribution you can attempt solving it explicitly (in every case when the mass distribution is homogenous, haven't tried it out otherwise) and find the inverse square law. As stated initially in this paragraph, this must not be. For example the earth's gravitational field is expanded in spherical harmonics for needs of expanded accuracy.
EDIT: Okay, too make that clearer: Yes, the lowest order approximation always gives you an inverse square dependence of the force of gravity. The point I mean here is that you cannot always say "The gravitational force is a square distance force from the center of mass" because that is only an approximation (albeit in most applications a good one - not always sufficient, though)
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