The process to becoming a navy physician (the marines are taken care of by navy physicians/corpsmen) is very similar to simply becoming a physician. The only real difference is if you do ROTC or Service Academy you have to apply through the navy to be allowed to go to medical school. There are two options for those who are ROTC to go to medical school: USUHS or HPSP. USUHS is the military's medical school in Bethesda, MD. It is run basically like a service academy where you are paid to go to school and wear a uniform. HPSP is the Health Professions Scholarship Program where you attend whatever medical school you are accepted to and the Navy pays your tuition and a stipend. You are in a reserve status during HPSP and active duty during USUHS.
On a whole to be allowed to go to medical school straight after college on a ROTC scholarship you need a strong application to medical school and would start working on this around your junior year of college. If you have any specific questions feel free to post and I'll try to answer.
LT, MC, USN