Would start out as a combined force taking volunteers based on what skills are needed regardless of branch of the military(to include coast gaurd) Eventually will be well enough established to be a fully independant service with direct recruiting and training
Ummmmm…I hate to break it to you, but you’re forgeting that the Army was the first, well, actually, the Minute Men were, but now every branch of the Military are basically Minute Men, ready at a moments notice (or rather were ready before there was a threat ;)).
Comin’ in hot…E.T.A. damn quick
The Air Force has a limited number of bases, and can project power anywhere because the world is relatively small at the speeds jets can travel at. Unless you can build lots and lots of bases, the Air Force model is not going to work. The Navy model is much better. (Maybe the Coast Guard model would work for local defense inside the solar system.)
ST:TOS used the 19th century British Navy model, with ships on long missions out of contact with home base. ST:TNG uses more of the modern model, with command looking over your shoulder in certain situations. They both seem to be fairly plausible.
If there were a network of Star Gates, the Air Force model might work better, since there is no reason to send people away from home for long periods of time.
It needs a new Flash Gordon-y rank structure…one might say a 1920’s style Death Armada…
Junior Space Scout (always named “Jimmy”)
Raygun Sergeant
Starsman
Isocorporal
Atom Chief
Laser Captain (must change name to an intimidating “Buck”, Flash", “Savage”, “Wildstar”, etc)
Space Yeoman (in miniskirt of course)
Nebula Marshall
I’m going to send in an orderly to, ah, “help” you take your meds, okay?
Please do not bite this time.
The people with the Nuclear Wessels should be in charge.
The correct answer is probably “none of the above”. Much in the same way the Air Force spun off from the Army Air Force in WWII, a futuristic space military would have very different requirements from either the Navy or Air Force.
The specifics would largely depend on what such futuristic speculative technology would look like. And based on the past 30 years of science fiction, we think it will probably look like this:
- Fleets of massive capital ships at least a mile in length armed with dozens of visually impressive energy weapons, missles and rail guns
- An onboard complement of manned attack craft that will swarm enemy behemoths as their motherships trade blows from visual range
- A garrison of dropship-mobile “space marines” of some sort. Usually armed with small arms and maybe powered armor if they are lucky.
- Possibly supporting armored vehicles - APCs, walking tanks, giant robots, etc.
- Maybe even exotic technology like invisible shields, tractor beams, teleporters, gates and so on.
-Conventional space warfare doctrines usually consist of hovering your ship a mile above an enemy city where you can either obliterate it at will or use it to wield political power the way only a million tons of space alloy floating above your head can. Or it consists of deploying your woefully ill equiped space-mobile forces to the planet in shuttles to be slaughtered by whatever enemy their weapons are ill equiped to fight.
Of course most of this will probably be wrong. To a certain extent, future anti-space ship weapons don’t have to be much more advanced than putting a rocket full of heavy ball bearings in orbit and letting them collide with their target at orbital velocities.
Actual laser weapons aren’t that far off and there is no stealth in space. So an invading ship has a good year or so of being cooked by a planetary laser to look forward to as it approaches an enemy planet.
Anyhow, I digress. You wouldn’t want to base your Space Force on the Navy or the Air Force because traveling in space isn’t anything like traveling on the ocean or flying in the atmosphere.
I would suggest some combination of Navy with NASA. I say NASA if for no other reason that space is a complex bitch and you will need men and women of dedicated science to tame it, if not sooner then definately later in your travels. NASA seems to have a good research platform to roll with.
Navy for most if not all ship to ship movement and coordination, plus any bomb dropping or missile launching. The idea of atmospheric warcraft (and thus, fighter pilots/air force) seems redundant based on the vast amount of potential atmospheres you could face in the huge wide universe. Atmospheric craft, IMHO, should focus purely on the task of moving men and material from space to a surface with as minimal compromise as possible (the problems alone with this task on many and probably varied planets alone is enough) and should heavily lean on the advice and analysis from any relevant geotechs, meteorologists or physicists etc. Any planetary bombing can happen from space with more power and less risk anyway. Marines, as mentioned, can provide boots on the ground.
Can we invent new rank titles?
Lieu-Tenant is so incredibly archaic.
Captain is the captain (person in charge) of the ship, not a rank.
General sounds so…general.
Private is likewise silly.
Major? Major what?
A private soldier, as opposed to one who has rank and privilege.
The major cause of your being busted back to private if you don’t get a grip and respect military discipline. 
I like lieutenant, major, general, and so forth, but I will agree that captain needs to be a position only, not a rank as well.