A mailing list which I'm on has discussed this.
The consensus was that which branch should handle it will depend on typical mission length. The navy has experience at handling multi week/multi month missions. On the other hand the air force is most skilled at multi-hour missions. Will the typical military space mission be a long time is space or a launch/couple of orbits/land operation?
Maybe had the Navy/Air Force capabilites combined into a single branch.. Hence the United States Space Force.