I actually hold a JAA Class 1, not FAA so I can't really answer your question. It was purely a cosmetic decision - I hated not being able to maintain proper eye contact with people and am delighted with the way it turned out. So I would have been issued a certificate regardless.
I had an alternating exotropia, so an eye always appeared to turn outwards. No idea why it's not mentioned in the FAR, but realistically it has no impact on your flying. There is a side-effect of a loss of stereopis which probably is mentioned (it's covered in the UK ATPL exams) but I never noticed a problem.
The only FAA medical I hold is class 3, and while the doctor issuing it was curious about the surgery, it had no impact on the medical. Although being Class 3, you could probably get away without having any eyes at all... ;-)