I don't think so. I have an idea for a counterexample but you will have to verify it for me because I'm not 100%.
The idea is that the modulus function is smooth, so if we smoothly stretch and apply a modulus function to the stretched space we should still have a smooth function, but not a spherically symmetric one (provided that our stretching doesn't preserve spheres!).
So here's what I propose. Let be any point on the unit sphere, and let for be the ray eminating from the origin through Then is the distance along the ray we have travelled from Any point can be expressed in the form and for any such point the function is a smooth, spherically symmetrical function . If we dilate that ray by a factor proportional to the altitude angle and consider the new distance from while leaving unchanged, then that distance would be a smooth function where but which is not spherically symmetric.
I guess what I'm saying is, for any in our domain, let for Expressing as in spherical coordinates, then the function has the desired properties but is not spherically symmetric.
This is more or less a sketch but if you can formalise it a bit, and if you can't find any technical problems with it, then hopefully it will work!