I think Kirk was referring to Starfleet being a combination of the Air Force, Navy, Army, Marines and Coast Guard.
Agreed, Kirk mentioned the "combined service" in response to Christopher bringing up "the navy."
Yes, I was unsure about the Antares because I couldn't remember if it was referred to in dialog as the "S.S." Antares, or if it was just the "Antares." I know for a fact it was never referred to as the "U.S.S." Antares, or as a starship. But yes, there is a possibility that it is Starfleet based on the crew uniforms.
FWIW, according to the memo Harvey
posted a while back, the behind-the-scenes intent was for Antares
to be a merchant vessel.