Wow, I saw a medical field thread in the USNA forum and thought for sure that I was going to open it and see the usual 'beat up the candidate who dares mention they would like to be a physician'; however, I must say that USNA1985 and Capt MJ have both provided some very good insight.
First as a little background of me, USMMA grad, sailed on oil tankers, then went to USUHS and will be pinning on wings as a naval flight surgeon in the near future.
The key if you want to attend USNA and be a physician is as already mentioned keep in the back of your mind that a likely result of attending USNA will be service as a line officer. If you are okay with spending a few (likely 5) years as a line officer before going to medical school then USNA is a great option for you and will set you up well for medical school no matter when you go.
Now, for some of the details that will be important later on, but will give you a good idea of how it will work at USNA. There is a limit to the number of grads who are allowed to go into the medical corps. The limit is, I believe, around 2%, which comes out to about 20 people. There is an application process at USNA which will take all those who are interested in the medical corps and decide who will be allowed to pursue that path. Now, if you are a competitive candidate for medical school acceptance (GPA ~3.5 or higher, MCAT >30, worthwhile extracurriculars, etc) then you will likely be competitive for the USNA nod. Along this path you will also submit your application to medical schools. You will likely apply to the military's medical school USUHS as well as any number of other "regular" medical schools.
At USUHS you would be on active duty as an Ensign and will be paid a full salary and benefits. You will not accrue any time until retirement (you will get the four years once you hit 20 though) and you will owe 7 years in addition to your 5 year USNA commitment (in reality the minimum time served will be 15 years total starting the day you graduate medical school). If you go to a "regular" college you will also apply to the HPSP program which is a scholarship program. In this program you will have your tuition and fees paid as well as receive a stipend each month. Currently the stipend is around 1400 I believe and there is also a sign up bonus, but I'm not sure if USNA grads get the bonus or not. After graduation from this program you would owe 4 years plus your 5 year USNA commitment (again, the real commitment would be more like 12 years minimum).
okay, slight pause for breathing......
Now onto the serve as a line officer for your initial commitment then medical school path. You would fulfill your USNA obligation as a line officer (had a handful in my med school who went this path, actually mostly former aviators) and then apply to medical school. Since your commitment to the Navy is up there is no board to apply to for the Navy to approve your path. You can again apply to both USUHS as well as the HPSP programs OR you can simply get out of the military completely and be a normal medical student who takes out loans. For USUHS you would accrue a 7yr commitment and for HPSP a 4yr commitment. This is actually a very viable option for many people and there are many "non-traditional" students in every medical school class. My class alone had three former naval aviators, one army armor officer, one navy seal, one army ranger, a handful of former enlisted, a former submarine officer, a vetenarian (yes, seriously), and I'm sure I'm missing a few others.
Okay, so what does it all boil down to: If you really want to be a physician USNA is a viable, but difficult option. If you really want to go to USNA then it is an awesome option. If you are okay with doing something else before being a physician then USNA is still an awesome option. If you want to be a physician no matter what then USNA is not a good option.