Am i sending out the wrong vibes ? does he think im infatuated with him?please help?

If he is showing you his wedding ring and talks about his wife, then he does think you are infatuated with him. However, if he is just overly using his left hand, who knows he might be ambidextrous. Most Dr. are very professional and unless you're wearing a thong that rides up when you sit down or you accidentally let it show when you are in a gown I don't think he is showing off his ring.