Some of the comments to my previous post have questioned the constitutionality under an originalist view of an independent Air Force. To my mind, the independence of the Air Force is not relevant to the constitutional question. If the Air Force is constitutional as part of the Navy, then it is also constitutional as an independent department.
To focus on the independence question, lets make the following assumption: The use of airplanes and other Air Force equipment would be constitutional if used by the Navy. That is, the term Navy in the Constitution does not preclude the use of this equipment. (This assumption must hold for the use of Air Force equipment to be constitutional as part of the Navy.)
Consider the following situation. Congress decides that instead of creating a single Department of the Navy, with a single Secretary of the Navy, it creates two departments: Navy Department A and Navy Department B. They are independent of one another, but both are under the control of the Secretary of Defense and the President. Would this be constitutional? Of course. There is nothing in the Constitution that requires a single department. While there is a single clause authorizing Congress to regulate interstate commerce, we have several agencies that regulate different parts of interstate commerce (such as the Federal Trade Commission and the Federal Communications Commission).
Now, add one more wrinkle: Congress has Navy A use different equipment than Navy B. This is also constitutional. There is no requirement that they be identical.
Finally, the last step: Congress changes the names from Navy A and Navy B to Navy and Air Force. This is obviously constitutional, since there is no requirement that any specific name be used. Put differently, that we call something the Air Force as a statutory matter does not decide the constitutional question of whether it is a Navy.
What this argument shows is that the independence of the Air Force is irrelevant. The key question is whether the equipment and weapons it uses are properly treated as part of the Navy or the Army. (Of course, there are other legitimate questions here, such as whether the Air Force should be included in the Army or the Navy. If the Air Force is properly classified as part of the Navy, then it is not subject to the two year appropriations limitation on the Army. But that is a separate question.)