I understand Navy Corpsmen serve as medics to the Marines.. who don't the Marines have their own medics?

The Commandant of the Marine Corps has his office in The Department of the Navy.  That means the Corps is a naval organization.  All Marine air assets are under a 3-star admiral responsible for  naval aviation.  As part of the USN (administratively) the Corps has no medical command/school, but Hospital Corpsmen are attached to Marine rifle squads (operationally).

 

Hope this info helped you out.