Those particles are virtual in that they don't really exist, so you are right that proving them isn't actually possible, because they are simply not there, just virtually, in our mathematical imagination. In quantum mechanics[1], this isn't really a "doesn't exist" kind of thing, rather it means that the wave function is there, leading to the (slim) possibility of existence through some kind of wave function collapse.

What is proven is that e.g. vacuum energy / zero point energy exists (not actually in the StarGate sense of extractable energy, just that the lowest energy state of any physical system isn't zero), and that the Casimir effect exists. Vacuum energy directly leads to virtual particles through pair production (which is a proven mechanism, at high energies, for low energies we do suspect that there isn't a cutoff there), and also influences e.g. high-energy cosmic rays leading to an observed high-energy cutoff (although there are other possible explanations for that cutoff and lack of very-high-energy cosmic rays). The Casimir effect is most easily explained by virtual particles and vaccum energy.

In Hawking radiation, the idea is actually that virtual particles through interaction with the gravity of the black hole become real particles. The event horizon actually makes those wave functions collapse such that real particles start to exist. Hawking radiation hasn't been observed yet, however.

[1] non-Kopenhagen QM has the same consequences, it's just even harder to explain actually.