In the Navy you can do a tour as a flight surgeon after your intern year. Its just a primary care doc for the airwing with training in "aviation medicine" but they also have to have a certain number of flight hours which is cool.
http://www.nasa.gov/audience/foreducators/stseducation/stories/Powers_Roden_Profile.html astronaut flight surgeon?
I think joining the military would be the best idea if you want to get out there!
I have been interested in doing military aerospace medicine and researched it fairly extensively. As a flight surgeon you are supposed to spend your time 50/50 as a primary doc and with flight RESPONSIBILITIES. These responsibilities include training, briefing, flying and other general prep. I think the minimum is something like 4 hours per month but often one can fly more. However, one thing to note is that you are never the pilot in command. you will always fly with a rated pilot who may or may not give you stick control while in the air.
as stated, after one year of interning you then are a flight surgeon. after a few years you enter a 3 year program giving you the aerospace medicine, an MPH, and a board certification in occupational medicine. however after completion of the 3 year program you are encouraged to move on from flight surgeon.
for one like me who would be happy as a flight surgeon but after military service would like a much different specialty, the military will rarely grant access to the flight surgeon program if you have already completed a residency in a different specialty; they would just put you in that specialty.
for one looking to fly more or specialize in something different, there are two other paths. you can either attend the pilot training program, "earn your wings" and then attend medical school. Your commitment length will be pretty long but you will be a pilot and a physician.
Another, more attractive offer for someone like me is to go through the reserves. they offer aerospace medicine. However, because the required time of service each year is minimal you can still pursue other things such as a civilian specialty. Likewise, you can train as a civilian physician and enter the reserves as a pilot so long as you complete the training course which i believe is 52 weeks; truly the best of both worlds.
oh and flight surgeons are a major path to becoming an astronaut, which is no promise of doing anything except training really hard to watch others go to space.
most of this information is from military forums and websites though as i have no direct experience with the programs at all.