No not really, the Navy, even when those in leadership werent aggressively supporting Naval Aviation, were still interested in keeping those assets 'in house' as it were.

The need for aviation to travel with the fleet, be it in the form of carriers, float planes from ships, or flying boats operating forward were all seen as integrated into the fleet itself. Indeed for a long time a young officer couldnt go to flight school right after commissioning and had to do a Division Officer tour somewhere else first.

That is not to say though that there werent central figures pushing expansion of Naval Aviation. The one of most notable of which is Admiral John Towers, Naval Aviator #3. In charge of organizing the Aeronautics Dept during WW1, he held a series of commands of increasing import between the wars. Including NAS San Diego, the carrier Saratoga and in 1939 was appointed the head of NAVAIR. He was the first in house aviator appointed to flag rank, and oversaw the organizing of the force during WW2, including being sent out to the Pacific in a shore based command.

What did give Army men like Mitchel fuel though was that the original Aeronautics Department set up in WW1 was actually dissolved for a time in the post WW1 drawdown by the Navy! Thus if they didnt care, why not follow the British route and consolidate all aircraft into one service to be managed centrally?

Also worth noting is Admiral William Moffett, the original chief of Naval Aviation in 1921, who fought many of the direct political battles vs Mitchell in the early 20's. He was backed by several influential political leaders too, notably the recently departed Assistant Secretary of the Navy, Franklin Roosevelt. Not an aviator himself, he still stuck with the fight and was able to argue that Naval needs were sufficiently different to remain separate. Moffett was also a main source of the decade of experimentation the Navy had with lighter than air and its small fleet of zeppelins. Sadly in 1933 Moffett was killed then in the crash of the airship Akron.

Now the Navy other senior men who were qualified Aviators than Towers and his generation, but that was part of a conscious plan by NAVAIR to take officers further on their career, teach them to fly, and use them to jumpstart their own advocates. Especially as command of carriers and Naval Air Stations were restricted to Naval Aviators. These 'Johnny Come Lately' officers were resented by some but produced some of the most important naval aviators of WW2 as well.

Ernest King(and Halsey) who followed Moffett as chief of NAVAIR was one of those men, and he had himself a strong ally in then now president FDR(as friendly a President as the Navy ever had). And from then on any thought of the Navy losing its aviation was put to bed. The one last incident being in the chaos of the late 40's post WW2 drawdown and consolidation efforts that created the modern Dept of Defense.