Just think of how hard it would be to prove a software random number generator is deterministic if all you have to look at is the output. Even computer algorithms like that, which are provably deterministic, observationally will still have some fluctuation in their output due to uncontrollable variables (corrupted ram, design limitations, hardware errors a la Intel floating point thing) and that's in an environment that is precisely engineered to produce such determinism. Then contrast that with all the millions of social, behavioral, and environmental factors that are throwing noise at observation of the human brain and it becomes mind-boggling very, very quickly.
Any possible proof of the determinism of the human brain would first require that we come to a complete understanding of the chemical and biological processes that control human thought in addition to how environmental and genetic factors influence those internal processes. I think this particular question will stay in the realm of philosophy rather than science for an extremely long time.