To be honest, at first I was upset with your question. I mean, every child in Poland is taught that Christopher Columbus discovered America. How would it be possible if America would be precisely synonymous with United States, as it was written by you. Also my Brasilian friend Rodrigo, who stays at my place for few days, asked me to write it here that he defines himself as an American.
But indeed, I cannot deny that there are plenties of situations where people use words such as America or American speaking particularly about United States and Polish language dictionaries mention it as one of common use of those words or even the main one.
I've started to check it by myself. There's no mention of Americans in Declaration of Independence nor the Constitution. But I've found it in few first inaugural speeches of US Presidents.
1st inaugural speech by George Washington
(...) but not without resorting once more to the benign Parent of the
Human Race in humble supplication that, since He has been pleased to
favor the American people with opportunities for deliberating in
perfect tranquillity (...)
2nd inaugural speech by George Washington
(...) When the occasion proper for it shall arrive, I shall endeavor to
express the high sense I entertain of this distinguished honor,
and of the confidence which has been reposed in me by the people
of united America. (...)
Inaugural speech by John Adams
When it was first perceived, in early times, that no middle course for
America remained between unlimited submission to a foreign legislature
and a total independence of its claims (...)
(...) In this dangerous crisis the people of America were not abandoned by
their usual good sense, presence of mind, resolution, or integrity. (...)
(...) Such is the amiable and interesting system of government (and such are
some of the abuses to which it may be exposed) which the people of
America have exhibited to the admiration and anxiety of the wise and
virtuous of all nations (...)
(...) if, while the conscious honor and integrity of the people of America
and the internal sentiment of their own power and energies must be
preserved (...)
(...) if an unshaken confidence in the honor, spirit, and resources of the
American people, on which I have so often hazarded my all and never
been deceived (...)
(...) With this great example before me, with the sense and spirit, the
faith and honor, the duty and interest, of the same American people
pledged to support the Constitution of the United States (...)
Finally, the first mention of the word "American" as a synonym to citizen of United States, I've found in:
2nd inaugural speech by Thomas Jefferson
(...) The remaining revenue on the consumption of foreign articles is paid
chiefly by those who can afford to add foreign luxuries to domestic
comforts, being collected on our seaboard and frontiers only, and
incorporated with the transactions of our mercantile citizens, it may
be the pleasure and the pride of an American to ask, What farmer,
what mechanic, what laborer ever sees a taxgatherer of the United
States? (...)
I hope it will help you in your researches.