Gauss' Law of gravity is:
$$\bigtriangledown \cdot \mathbf{g}= 4\pi G\rho$$
This can be shown to be equivalent to Newton's Law of gravity via the divergence theorem. However, this does not really constitute a proof. Where does the $4\pi$ come from? I would like to derive Gauss' Law from the notion of solid angle and/or the definition of the scalar potential.
Specifically, just using the following facts:
$$\mathbf{g}=\bigtriangledown\cdot\phi$$
Where $\phi$ is the scalar potential, and the definition of the scalar potential (from the wikipedia article):
$$4\pi\phi=\int_V \frac{\bigtriangledown\cdot\mathbf{g}}{r}\:\:dV$$
(Where the RHS is a volume integral, and the necessary assumptions about asymptotic vanishing towards infinity are made).