The main problem as I see it, is the definition of "this coin".
You can prove, for reasonable definitions of "prove", that "this coin" is fair, by tossing it enough times and showing it has a 50/50 outcome with a reasonable confidence interval. (Or prove that it is unfair by tossing it and showing that with a reasonable confidence interval, the coin does not have a 50/50 outcome). But "this coin" will in that case mean only the coin which you tossed as it was when you tossed it. It doesn't show that it will continue to be fair, since somebody can tamper with the coin, in which case it stops being fair.
But in normal parlance, we would still call it "this coin", although from the scientific/statistical/philosophical standpoint it is no longer the same coin once it has been tampered with.
In other words, you can only prove that the coin was fair, not that it will be fair.