No, the inverse square law is not an assumption. It is a version of Gauss' law. By locality any force is communicated by "force carriers" . As they emerge from a spherically symmetric source they will loose their intensity by an amount which scales as the area of the sphere surrounding the source. eg. in two dimensions the ripples on a pond will have an amplitude that decay as 1/(2 pi r). In three dimensions the the surface of a sphere is $4 pi r^2$ thus any force in 3d will decay by $1/( 4 pi r^2)$ and in D dimensions by $1 / r^(D-1)$ with constant of proportionality related to the area of a sphere in that dimension.