The basic requirement for a new branch of the military is that it can accomplish military objectives independently and in a new domain that is not covered by existing branches.
The US Navy (and I suspect others as well) already does have a huge air force. There is no sense in having the Air Force operate out of aircraft carriers, since those aircraft have the primary goal of winning naval engagements, so the Navy may as well take over there. Note that this is different even from the Marine Corps, which works closely with the Navy but handles marine engagements. They are not generally involved in naval battles.
In the land-based case, the Army's "air force" was originally used mostly for terror bombing and, more broadly, for attacking infastructure that the Army could not reach. It's primary goals where, therefore, separate from the Army's goal, and hence you get the Air Force breaking off from the Army while the Navy still has it's own "air force."
To finally answer your question, all you really would have needed to keep aircraft at sea exclusively is for the WW2 era terror bombings to be ineffective or illegal. There were actually discussions between representatives of the major powers in the 19th century about making terror bombings illegal on a moral basis, so the latter option is less implausible than it may seem at first. Moreover, prominent post-WWI military thinkers actually thought that terror bombing would be so effective that civilians would basically force their government's hand in surrendering almost immediately. This was, in fact, one of their moral justifications for the bombings (war would be brutal, but at least it would be short!). 527
The fact that civilians could sustain years on end of bombing, sometimes while sustaining industrial output, was a big disappointment. I could have seen this leading to bombings to be outlawed as needlessly cruel for little effect. In the US, I think it's well known that the effectiveness of bombings in Vietnam were also a huge disappointment. Now that I think about it, it's sort of weird that such tactics didn't fall out of favor.
Anyway, if the Navy is the only branch with air power, I think you pretty much accomplish your goal. I know I didn't mention the Marine Corps but ... it's really hard to imagine nations not needing traditional land armies. For a lot of history, that's the only branch most nation's had.
Edit: You would very likely take interest in an episode of the Hardcore History podcast titled BLITZ Logical Insanity.
Edit 2: <PERSON> is contesting (in the comments) whether American's participated in terror bombings or only in precision raids. Rather than argue the point further, I will just point out that if you agree with @RonJohn, then you can swap out the Americans for the British in my example (as they point out) and my basic points still hold. The distinction between precision and terror bombings isn't actually that relevant to the question anyway. 347