Flight surgeons go through an abbreviated syllabus in T-34s, but do not qualify as naval aviators. They do get flight pay, providing they get in the air each month in some manner (like our Doc, who goes on all our road trips with us. Comes in handy in places like Vegas). But for the most part, Docs don't fly or log time.
Of course, there's an exception to everything, I went through primary with an O-4 flight doc who somehow go the navy to put him through the entire syllabus. Also, I don't know if he's still there, but the flight surgeon at VP-30 in Jax when I went through the FRS was the crustiest old jet dude I've ever met. I don't know if he was a pilot first and then went to the medical corps or what, but he wore both Doc and Pilot wings on his nametag.