There's one possibility, but let's explain why it's impossible first
About 1/3 of Earth's surface is land. Land has incredible advantages...
- Fuel & maintenance delivery is cheap.
- You can't sink a continent.
- You're next to the manufacturers.
- You're next to the training grounds.
- Landing at night is nowhere near as big an issue.
- Plane size is limited by the physics of the plane, not the physics of the ship.
And I could go on. Land has huge advantages when it comes to almost everything. On the other hand, your navy...
- Has more trouble launching planes in bad weather than land-based operations ever would.
- Must have raw material deliveries for fuel.
- Is susceptible to torpedoes.
- Is limited in the number of planes it can field (compared to comparatively infinite land).
- Will always limit the size of the plane.
And, above all...
- A navy is expensive. Really, really expensive.
If they must, planes can use a field to land and take off. Compare the cost of building an airport (about \$30 Million / 3km air strip) to the cost of the U.S.S. Gerald R. Ford (about \$12.8 Billion, ignoring the nearly \$5 billion research price tag).
Frankly, the economics of the issue alone kill the deal. Aircraft carriers would have to be cheap as proverbial water, which isn't believable. And that's ignoring all the other pros and cons. It's definitely useful to have a mobile air force (air craft carriers) such that one needn't negotiate landing privileges all over the globe — but it's a long way away from never having an independent air force.
Having said that... I did think of one possibility
Historically, small rockets (aka fireworks and limited munitions) came first, then airplanes, then big rockets (V1/V2, etc.). By the time ICBM missiles came along, the Air Force was already established and it made logical sense to put the two flying things together in the same military unit.
But let's keep in mind that the U.S. air force (to use as my test example) started as the Army Air Corps. If we assume that rockets came first and practical airplanes second, then we might have something. And I think that's a reasonable divergence from history (frankly, given that the Chinese were using rockets hundreds of years before the Wright Brothers, it's a wonder that practical rocketry didn't come first. The mathematics of ballistics aren't that difficult, but I'm sure Werner von Braun could give me a sound whoopin' over that calloused and uneducated statement.)
Anyway, practical rocketry would make a ton of sense in the navy. Think about it, the allies used battleships to soften the beaches at Normandy during WWII before the D-Day invasion. Practical rocketry would have provided (I believe) more payload over a longer distance, making that effort much more efficient. In fact, the navy would become the primary mid-to-long distance bombing solution rather than airplanes (which didn't come until later in this new timeline).
To be fair, this suggestion would be hard-pressed to overcome the other advantages of a land-based air force. But if we let that slide for a moment, there would never have been an Army Air Corps. It would have been the Navy Missile Service (NMS), and the burgeoning air force would have been absorbed into the navy, only receiving a separation (as it did with the Army) when intercontinental plane designs necessitated abandoning ship (think B-29 Superfortress and bigger).
I don't think it's possible to justify air support staying with the navy. In other words, it would eventually split into two services anyway. But by then the "Air Force" would be modeled after the navy, not the army.