Hey here's what I can tell you,
If you want to join the Navy as enlisted, you can do that if you have just your green card, no biggie and when you've fulfilled five years of legal residence, you will get your citizenship pretty quickly especially if youre a vet or still in the military.
If you want to go in as an doctor (officer), you have to pass your boards for General Medical Practice here in the US first and foremost. If you pass them to become a naval officer, you must be a citizen, but since they are in shortage for physicians at all departments of the military, the Dept of the Navy may be able to pull some strings and do some kind of citizenship in exchange for a certain year commitment, but really and truly, the boards are the ones that get you.
I used to live in Mexico and I was studying medicine before I decided to change my major to international business and I knew if I ever wanted to practice med in the US, I would have to pass the boards and those are tough ass. GOOD LUCK!