I think culture will need to adapt so much that the current US military branches will not be as relevant as this question supposes.
Notice that when aircraft became relevant to the US military, they were organized underneath the Army, Navy and later Marines as separate elements. My details may not be entirely accurate, but roughly, then the Army Air Force became the USAF, and the other branches retained their own air components. In Britain, aircraft started out organized under the army cavalry, before the RFC (Royal Flying Corps) and then RAF were formed. Etc.
Currently of course, and hopefully as far as possible into the future, the main space effort would be non-military and thus in the US mainly NASA. Though there’s also JPL and private companies. More and more nations and companies are fielding efforts currently.
But there are mountains of shit liable to hit various fans before there’s any stable space fleet. The USA and its military structure will be much different, if they even exist, by that point.
Also there are many many assumptions to fill in about technology, missions, and areas of operation and so on, which would all inform an answer to such a question.
In general though, I think spacecraft with limited crew and missions such as we will have for a long time, more closely fits an Air Force metaphor than a naval one. But (assuming this happens at all) once, as your question asks, space ships are fairly common and are large and capable of independent and long-term operations in multiple roles, with entire staffs and command hierarchy aboard each, that seems to more fit a Naval metaphor.
But hopefully (necessarily?) if we’ve survived and advanced technology that long, we will also advance our culture and thinking so that there will be little/no need for a military organization, because we will not be having conflicts that call for them. If people have their physical, emotional and psychological needs well met, they won’t be inclined to do such violence, and crazy people will be detected and healed before they get anywhere near control of space technology.
In that more optimistic/peaceful case, I imagine there would be different organizations for different purposes for the use of space ships. Perhaps there would be some force-capable elements that would be needed in case of rogues, but I’d think those would actually follow more of a police / investigation / ranger / Coast Guard model/metaphor than a military one.