So the aircraft carrier will be operated by the Navy, but the aircraft might be Marine aviation aircraft, such as in the case of the F-18 used by the Marines. The Airforce operates no carrier based aircraft. All US carrier based aircraft belong to either the Navy, Marines, or Army.
Although Marines often are deployed on ships, they aren’t stationed on ships. Duty stations are Marine or Navy bases that are home to the Marines between deployments, usually for a year or more.