I was listening to an audio book today "The Colonels" by WEB Griffin and in the story they were reading off some Confidential operations orders. In the orders they were transferring some US Navy planes to the US Army special forces for a mission to Cuba. They were to remove the Navy insignia, but no where was the aircraft to be labeled as belonging to the Army. The orders went on to say that the only markings would be the tail number and USA on the lower side of the left wing and USA on the upper side of the right wing.
In my years in the USAF I knew the military branches to be labeled as:
In these names there is nothing mentioning "of America". Yet the lettering on the Aircraft, in the story, was to be USA.
This got my mind twisted. My whole life I've been fascinated with the military. My family has been in one branch or another for as long as we know; going back to serving in Ireland. I have known many military acronyms since childhood. I can't remember a single time I've seen anything labeled with USAA for the Army and the same for the other branches. Even naval ships use U.S.S. for United States Ship. I also know the Constitution is known as the U.S. Constitution and doesn't have the "of America". I know it reads "We the People of the United States" and paragraph one ends with "establish this Constitution for the United States of America". One lists "of America" and one doesn't.
WTF! Have I been calling my home the U.S.A. mistakenly? Should it only be U.S.? Is the "of America" just to let people know which "United States" country we are talking about (maybe there are other countries call U.S.)?
I know deep down that we are called the USA, but my mind is having trouble wrapping itself around why we always leave off the "A" on our military and government buildings, machinery, weapons, and documentation.
(This may need to go to /r/Military but I come to this sub daily and I figured I would try here first)
The Declaration of Independence only makes reference to the 'united States of America', never to just 'the United States' (referring to the union of various independent or quasi-independent states before federation proper); the Constitution explicitly states that it "[does] ordain and establish this Constitution for the United States of America". Likewise, the President is explicitly defined as the "President of the United States of America", and its ratification was affirmed with the text "DONE in Convention by the Unanimous Consent of the States present the Seventeenth Day of September in the Year of our Lord one thousand seven hundred and Eighty seven and of the Independence of the United States of America".
The 'United States of America' is today the official name of the nation-state conceived by the Constitution; the Constitution is explicitly defined as creating a 'United States of America' (with full capitalisation, unlike the earlier Declaration of Independence). The use of 'United States' is simply a short-form name, a reduction of 'United States of America', for the sake of simplicity and the avoidance of redundancy. Most nations make use of such a short form; for instance, many legal documents in countries with the name 'The Republic of X' will refer to themselves alternatively as 'X' or as 'the Republic'.
Take the UK, for instance. Its legal identity is the United Kingdom of Great Britain and Northern Ireland; the acronmyn UKGBNI is never used, however. Instead, the name is shortened to just the United Kingdom or the UK - there isn't much more to that than convenience. As it happens, no other modern nation carries the descriptor United States or United Kingdom (Mexico is the United Mexican States), which makes both convenient shorthand internationally as well. Most nations of the world have a formal name including a descriptor (like Republic, Federation or Kingdom) and a 'proper' name (like Britain, France or America) for the country or the people (East Germany, for instance, was the German Democratic Republic rather than the Democratic Republic of Germany). With regards to the military, the US is not alone in using short form names. Australia's armed forces are the Australian Defence Force; not the Commonwealth of Australia Defence Force, as they would be if they took the legal name of Australia.
You are equally right to refer to the nation as either the United States of America or just as the United States. In an international legal context, the United States of America would be the correct form of use in the first instance (and may thereafter be reduced to United States), whilst within the US either would be acceptable. It is common in general for legal documents to refer in the first instance to a state's full name once and thereafter reduce it. Take for instance the Convention of 1818; it is explicitly negotiated on behalf of "The United States of America", but thereafter the document just refers to "The United States". The 1858 Harris Treaty with Japan also opens with "United States of America" and thereafter just says "United States" or "Americans".
It's easiest to think of "The United States of America" as the proper full name and "United States" as the proper adjective - though the latter is fine too as a short form of the name.
There have been other "United States of"s - Brazil, Colombia, Venezuela - all since renamed "Republic of".