The final and undeniable proof that there cannot be a classical physics explanation of quantum mechanics will be when a real quantum computer has been constructed and is able to factor a huge number that current classical computers cannot factor in a time less than the age of the universe.  This demonstration of quantum mechanics will be far more convincing than the other interference and entanglement experiments and no-go theorems that Christoph mentions in his answer:

> ...there's a fundamental disconnect between quantum and classical
> theories and various no-go theorems that go along with it (Bell,
> Kochen-Specker, Greenberger–Horne–Zeilinger are probably the most
> famous ones).

On the other hand, if we are unable to build a quantum computer or if they do not work we will know that there is something fundamentally wrong with quantum mechanics and will have the fun of finding the new theory that explains why quantum computers won't work and still gives all the other well tested results that quantum mechanics predicts.