I would think that the Airforce would be the natural father of a space force. The beginning space fighter/bombers will likely be like an oversized version of spaceshipone which could use a vertical rocket as it's first stage. It would have to launch from land, the Navy might have a claim when we build space-only battle cruisers, but that's not going to happen for a long time.
It has to be Navy. The clue is in the name "Air" Force.