Would calculating the nth prime ruin encryption? I'm sure there is some proof that the nth prime can be found. But if we knew, would encryption that relies on primes be easy to decrypt?
No.  The only plausibly relevant major cryptosystems involving that are RSA-based cryptosystems.  A typical RSA prime is chosen uniformly at random from 1024-bit primes, of which there are $\pi(2^{1024}) - \pi(2^{1023}) \approx 2^{1014} - 2^{1013} = 2^{1013}$ possibilities, by the prime-counting approximation $\pi(x) \approx x/\!\log x$.  Even if you could efficiently find the 1762496654486650458301017412708452577168455321425338564295164204627851827809673950656701400197184589192372578810839228796442397889449007625305174377608286256232618793314981228756675796669238641660676263338471020306341778397911556529236704830386308649499298417719554769934355960770722959180457300530690137th one, for example, that wouldn't help you to guess which prime I picked.  You don't even have a hope of guessing which of the $2^{256}$ ChaCha keys I'm using, and there are a lot fewer of those.