I'd say it is. Here is my intuition at a proof.

Imagine that we shift the sphere continuously along any direction.

During this shift, any point for which f increases must be matched with another point for which f decreases commensurably.

Now take the circle of points where the associated spoke is perpendicular to the direction of travel.

By adding an infinitesimal scalar product of the vector in the direction of travel to the spoke, in negative and positive terms, two more circles are produced, infinitesimally shiftsd from each the original circle in the direction of travel

After the shift is applied, one circle will switch positions to the position of the next, and the other circle will shift positions still.

Now there are two possibilities for the integral not to change. Either the function is infinitely symmetrical, which is only true of constant functions, in which case the function must be null.

Or, the change of the values of the two circles must be constant, which means that the derivative of the integral of a circle over any direction must be zero. By repeating the same process with spokes, you come to find that either there is infinite symmetry or the derivative of the integral for a point is constant, in which case the function is constant.

In conclusion, the function must be constant. The only constant function for which the surface integral of a sphere is zero is the null function, therefore f is the null function.

I'm sure I'm being unrigorous somewhere or missing something, didn't take multivariate calculus yet. But this is my intuition at an answer :)