[lmm]

The standard answer is the Bell inequalities, but if you only care about deterministic vs. nondeterministic then Conway's "free will theorem" is a bit simpler.

(The result is that either which quantities we choose to measure must be deterministic, or physics must be nonlocal (i.e. what we do in one place can change what we measure in another place even if the two experiments happen simultaneously), or the results of our measurements must be nondeterministic)

[Retric]

Nothing says we live in a local universe.

[lmm]

There's no proof, sure. But nonlocality is very counterintuitive, arguably more so than nondeterminism (and there are no familiar classical/macroscopic phenomena that exhibit nonlocality, whereas many familiar things at least appear nondeterministic in practice).