To answer your orginal question, first off, as a marine you cant become a medical person in general. Like previously stated, all the tstaff corps and support comes from the navy. A corpsmen is an enlisted job, so as an officer you would have to becoem a nurse or a doctor or a PA, or something along those lines.
so to do what you wanted, you would have to laterally transfer to the navy. But once you would do that, its hard to become one. the waiting list and the need is long and high. The school is realitivly short compared to the things that they can do, but there are so many other little courses and classes that they have to take. They are the most decorated corps in the Navy, they are veyr protective of who gets in and such.
Good luck, hope that helps.