By the end of the first half of the 20th century as Airpower became entrenched as the Third Dimension of warfare countries increasingly made their Air Arms into separate services rather than component branches of land forces. The RAF was formed from the Royal Flying Corps and Royal Naval Air Service, but Fleet Air Arm remained. The USAAF became the USAF, but Naval Aviation remained. Same with French and Soviet Air Forces.
Why has the trend been this way?
Looking specifically at Britain, the Fleet Air Arm didn't remain after the formation of the Royal Air Force in 1918; the RAF entirely subsumed the aircraft of the RFC and RNAS. In theory the new air service should have remained "in intimate touch with the Army and Navy by the closest liaison" according to the Smuts Report of 1917 that had recommended the formation of the RAF, but as the armed forces dramatically shrunk after the First World War and defence spending was slashed the three services squabbled incessantly over funding and control. Navy aircraft carriers embarked Air Force aircraft and pilots, an awkward situation that persisted with modifications during the inter-war years. The Fleet Air Arm was formed in 1924 as The Fleet Air Arm of the Royal Air Force, with both RAF and RN personnel, naval aviators holding dual ranks in both services, still an awkward situation only marginally more acceptable to the Navy who continued to (quite justifiably) press for full control that they achieved in 1939 following the decision of Sir Thomas Inksip, Minister for the Co-ordination of Defence in 1937 ("The Fleet Air Arm not only co-operates with the Fleet; it is an integral part of the Fleet").
The state of the Fleet Air Arm at the outbreak of the Second World War illustrates some of the problems of independent air forces (though every country had its own situations and challenges). Maritime aviation was far down the RAF's list of priorities, and therefore expenditure, and the process of aircraft procurement became even more Byzantine when factoring in Navy requirements; limited numbers of aircraft meant there was pressure to combine multiple roles, sometimes successfully (the Fairey Swordfish torpedo/spotter/reconnaissance, highly effective despite being a biplane) usually less so (e.g. the Blackburn Skua fighter/dive bomber, not a success in either role). Attempts to plug the gaps with land-based fighters converted for carriers (e.g. the Sea Gladiator, Sea Hurricane and Seafire) were far from ideal; the Seafire had the performance sorely lacking in previous fighters but was short ranged and difficult to operate from carriers.
Whether the experiences of the RAF and FAA were taken into account when the USAF became a separate service I couldn't say, but aircraft operating from carriers present distinct challenges of operation and control that have generally been best met by one service (where budgets allow), rather than relying on the co-operation of two.