Everyone makes good points here, especially the navy vs air force thing. The air force, being airborne and requiring fuel to stay there, is not a long-term-deployment service. The navy, however, is. You don't need to use fuel to stay at sea (you're likely to never get back to land if you run out, actually), nor do you need to use fuel to stay in space (excepting unstable orbits).
The space force splitting off and becoming more navy-like over time is something I would expect to happen. This would not really happen, though, until we had viable bases in space already, to start from and head to. The air force would certainly start this, as fighters etc need to launch/land from *somewhere*, and doing so from Earth's surface would be *very* impractical.
So. Initially, the air force. As our spaceborne infrastructure develops more and more, more like the navy.