All of our models of reality are restricted to computable functions. However, we know that uncomputable functions not only exist, but that nearly all functions are in fact uncomputable. Therefore, it's well within the realm of plausibility that the actual behavior of the universe is governed by uncomputable functions, and we are forever stuck modeling those behaviors with computable approximations.
To claim that reality is entirely computable, one has to show how uncomputable functions can exist therein. I wouldn't call this proof, but it strongly suggests to me that the behavior of the system we call reality is uncomputable, and that subsystems thereof also may not be. If what we call human intelligence is one of those uncomputable subsystems, then it's true that computational AI will never achieve it. Nonetheless, we've gotten pretty far with computable approximations, so machine "intelligence" that's close enough for practical purposes doesn't strike me as impossible even if we inhabit an uncomputable reality.