I'm a pilot in the Army, I am in the process of finishing my pre-req's for med school, and I intend to go into aviation medicine. I new that if I wanted to get some real aviation experience I would have to go aviation first. Getting your private license is NOT the same as being trained by the military. Over 75% of the people I went through flight school with had their private license, many had degrees in aerospace and it offered almost no advantage over someone with no experience, especially after the first month. We had a guy with 1200 civilian hours and everyone caught up to him within a month or two. The rules are more lax in civilian aviation and so is the knowledge retention and application.
Our flight surgeons don't fly, it's not their job, they do get exposed to it, but not nearly enough to get you anywhere in aviation or make you comfortable enough to actually fly an aircraft, as in takeoff, pattern, and land.
If you really want to be a pilot and a doctor think about how important flying is to you. I say this to everyone that tells me they want to go to flight school because it's a pretty intense curriculum, and if you don't want it bad there will be far too many times where you want to quit, especially in SERE school, and primary. (we had a bunch quit) If you have to do it, consider flight school first. The skils you will learn will set you apart in the civilian world and only help you on a med school application.
I had a doctor in my flight school class, he told me that he got bored with it, and just had to learn to fly helicopters. I was surprised because I asked why he didn't just go and get his private license, but he said he wanted to get the experience that you get in the military (million dollar flight training and crazy amounts of flight hours). I asked him how med school was compared to flight school, he said it was about the same except there were times in flight school that were far more stressful and time consuming than he experienced in med school. I'm sure that depends on where you go though.