USNA '16 - just noting your opening comment:
I have always been interested in medicine, but understand that all SA have a priority of first producing officers. USNA's main mission is to produce warfare officers. Navy nurses are most definitely naval officers, just not warfare. They are staff officers.
I am not aware of any direct to USNA-to-Nurse Corps options, unlike the Medical and Dental Corps options.
The Navy gets most of its nurses through direct commissioning programs after graduation from a civilian college or university with a BSN program. Check out the very generous Health Professional Scholarship program at this link:
http://www.navy.com/careers/healthcare/nurse.html
As noted by other posters, there is the NROTC Nurse Corps option, easily searched for by using "NROTC Nurse Corps option."
Take a look at this typical BSN university website; I think it's a good example of a typical program.
http://web.usf.edu/nrotc/nrotc/Nurse-Corps.htm
The long way to do this is attend USNA, enter a warfare community, leave active duty after obligated service time and use GI Bill/veterans' educational benefits to return to school for BSN, then ask to come back in as a Navy Nurse. No guarantees. Though there are lateral transfer boards allowing for transfer between one officer community into another, I don't think that is the norm for Nurse Corps. I am sure there are exceptions to the rule out there, who have somehow done their "day job" as a warfare officer and completed required courses during off-duty hours to progress toward a BSN and ...but that would be very rare.
For even more down-the-road knowledge, check out the Uniformed Services University of the Health Sciences (USUHS) Graduate School of Nursing. Military healthcare providers have a long and storied history of serving, and a career in Navy Nurse Corps can be long and rich.
http://www.usuhs.mil/gsn/index.html
If you really want to be a naval officer in the Nurse Corps (Army and Air Force have similar programs), the most direct path does not lie through USNA. Remember USNA is not an end goal in itself, but an interim and very powerful launching pad for a minimum five-year career as a warfare officer.
It's good to think about all the options. The more you educate yourself about all the options, the better able you will be to choose. One of the hardest things at your age is choosing among so many interesting paths. Good luck!