There's a good 60 Minutes segment on this from a few years back: https://www.youtube.com/watch?v=6mXl8C4-M_4 I also believe you have to follow the Greek Orthodox religion to be able to visit - they might make some rare exceptions. (might be wrong)
You need a permit (effectively, a visa) to enter the peninsula, and there is a daily quota. About a tenth of available permits can be issued to non-Orthodox visitors. They don't single out the Greek church from other Orthodox churches (AFAIK).
[I researched the entry requirements a couple of years ago out of curiosity, but never tried to actually visit. Something may have changed in the meantime, but somehow I doubt it :)]