Shinto priests marry. I'm not sure about buddhist ones and he is reciting sutras, which would imply he is one.
That being said, I have suspicion the whole "no-girls allowed" thing has more to do with the fact he has no mom rather than being monk.
Looked it up, and contrary to normal Buddhist monks, those in Japan DO get married and have kids so they have a new custodian in line to take over the temple. What then doesn't make sense is how's Medaka supposed to produce an heir if he can't interact with girls in the first place?
Inb4 there's no actual rule about not getting close to girls, and Medaka just really misunderstood something. It'd be hilarious if it got to the point where he was so in live with her that he took her home and was willing to defy his family, only for his family to be like "bro, tf are you talking about? We never said to avoid girls completely."