We've had to unfortunately remove all comments because none of them were even close to being appropriate for this subreddit. Remember the rules. As was explained in the excellent recent meta post on "What it means to post a good answer in /r/AskHistorians":

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This often seems to be a major point of contention for non-Americans, but I really think it's just a misunderstanding. It's a tough question to address because of the passion the question sometimes draws.

However it is important to note that there is a definite distinction between the continent names, North America and South America, and a country with "America" in its name, as in United States of America. Grammatically this means that the country is America and it is formed by a conglomeration of States.

Here is another way to look at it:

The official name of Mexico is "United Mexican States" hence citizens being called "Mexicans."

If you're inquiring about the history of the term, the Oxford English Dictionary does note than American was used by Europeans initially to refer to indigenous populations. By the middle of the following century however, its use had been extended to colonists. It actually seems that Europeans were referring to the colonists as Americans before the colonists themselves were.

Post revolution, there was a big push among leaders for citizens to think and refer themselves as Americans. This was an effort to bring the nation together, as many people still thought of themselves as Virginians, Pennsylvanians, or otherwise.

Lastly, I'm just not sure what else we'd call someone from the United States of America!

If you mean the average American I'd argue it was the War of 1812 that first indicated the shift. As stated in Ian W. Toll's Six Frigates, "it was only after the War of 1812 that Americans began speaking of the United States in the singular rather than in the plural." It was the second war of independence for many. This shift was mostly complete by the time of the Monroe Doctrine in 1823 as it made it an American (U.S.) continent. Daniel Walker Howe's What Hath God Wrought describes this and Clay's "American System" in great detail. This would be cemented by the U.S. victory in the Mexican-American War (1846-1848, also in Howe's book) but of course the union would be contested in the American Civil War (1861-1865) and perhaps not globally accepted until the hegemonic Roosevelt Corollary in 1904.

When looking at these sources and time periods it's important to note that the British called the U.S. citizens "Americans" even earlier than this and it was first noted locally after the First Barbary War (although Six Frigates covers both of these). Public figures speaking for their nation don't really count because the vast majority of people lived rurally and were disconnected from greater events until the development of the telegraph (Howe's book covers this in good detail). Really, I wouldn't call it a (locally) universal term until the 2nd half of the 19th century due to technological, demographical, and political changes, but the essential criteria of your question would have been met between 1815 and 1848.

Google's Ngram viewer shows the first mention of 'Americans' around 1750 though the more dead ringer term for this sort of thinking ('American citizens') only begins in the early 1800's.

The Civil War appears to cement this, though, and/or possibly the Alaska purchase.

I find it neat my page search here did not bring up anyone mentioning the Monroe Doctrine. Can anyone explain why as it essentially reinforced a manifest dynasty of all the Americas?