As a point of clarity, "United States" was added to "America", not the other way around.

Early uses of America specifically referencing what later became the USA exist from colonial time. For example, when Benjamin Franklin offered his Albany Plan of Union in July of 1754, it included the term "America";

Plan of a Proposed Union of the Several Colonies of Masachusets-bay, New Hampshire, Coneticut, Rhode Island, New York, New Jerseys, Pensilvania, Maryland, Virginia, North Carolina, and South Carolina,9 For their Mutual Defence and Security, and for Extending the British Settlements in North America.

That humble Application be made for an Act of the Parliament of Great Britain, by Virtue of which, one General Government may be formed in America, including all the said Colonies, within and under which Government, each Colony may retain its present Constitution, except in the Particulars wherein a Change may be directed by the said Act, as hereafter follows...*

He literally meant "British North America" by America. This wasn't even the first time Franklin used the term;

Know all Men by these Presents, That We Benjamin Franklin Deputy-Postmaster of the City of Philada. in the Province of Pennsilvania, and Robert Grace of the same Place Merchant are Held and firmly Bound unto the Honourable Elliott Benger, Esqr; Sole Deputy-Postmaster General of all his Majesty’s Dominions in America... Franklin and Grace to Benger, Apr 1744

Here he's a bit more specific in saying "his Majesty's Dominions in America," but the place is America and the specificity is to his Majesty's Dominions there. So he is again calling the land "America." But this wasn't the first time.

Alexander Spotswood was larger than life. He dispatched Maynard to capture and kill Blackbeard the pirate, he "found" the Shenandoah Valley, he was instrumental in introducing dancing and balls to Virginia elites, and he served as a very early governor, being the first to reside in the governors palace at Williamsburg (technically, though, he was merely a Lt Governor, but the actual governor never visited Virginia). He was also appointed a position Franklin would later hold - Deputy Postmaster General of North America - and from 1730-1739. As such he had to deal with local postmasters. In 1737 Franklin became postmaster for Philly, which gives us;

I am obliged every half Year to have the Accounts of the General Post-Office in America made up: But I have not been able to obtain any Account from Mr. Bradford of the Philadelphia Office, from Mid-summer 1734, notwithstanding all the pressing Demands that the Comptroller has been continually making upon him for so many Years past. Spotswood to Franklin, Oct 1739

It goes back much further. William Penn (1683);

XXVI. For their Original, I am ready to believe them of the Jewish Race, I mean, of the stock of the Ten Tribes, and that for the following Reasons; first, They were to go to a Land not planted or known, which to be sure Asia and Africa were, if not Europe; and he that intended that extraordinary Judgment upon them, might make the Passage not uneasie to them, as it is not imposible in it self, from the Easter-most parts of Asia, to the Wester-most of America.

He meant the continent, obviously, but the title of the letter is A letter from William Penn, poprietary and governour of Pennsylvania in America, to the Committee of the Free Society of Traders of that province residing in London containing a general description of the said province, its soil, air, water, seasons, and produce... (actual the title keeps going a bit).

This was 1683. But 100 years earlier (Apr 1584);

The First Voyage to Roanoke. 1584. The First Voyage Made to the Coasts of America, with Two Barks, wherein Were Captains M. Philip Amadas and M. Arthur Barlowe, Who Discovered Part of the Countrey Now Called Virginia, anno 1584. Written by One of the Said Captaines, and Sent to Sir Walter Ralegh, Knight, at Whose Charge and Direction, the Said Voyage Was Set Forth:

If you saw my post on the Lost Colony of Roanoke a couple weeks ago you may remember Capt.'s Amadas and Barlowe "found" Roanoke Island, which Raleigh named Virginia (which of course included much more than just the Island of Roanoke). It was on their recommendation that the first colony was settled there, unbeknownst to them dooming the subsequent efforts to colonize by Raleigh in America. But that wasn't even the first time the future colonial attempt called it America, which came from the queen herself in Raleighs Charter a month earlier (Mar 1584);

CHARTER IN FAVOR OF SIR WALTER RALEGH, KNIGHT, FOR THE DISCOVERY AND PLANTING OF NEW LANDS IN AMERICA, 25 MARCH 1584.

So you get the point that Anglo culture referred to the land as America since before they ever showed up. This can keep getting kicked back to the map of the early 1500s that first named the place America (the Waldseemüller map, 1507).

When did it get specifically attached to the USA? 1776. It was all up in the Articles of Association (Oct 1774);

We, his majesty's most loyal subjects, the delegates of the several colonies of New-Hampshire, Massachusetts-Bay, Rhode-Island, Connecticut, New-York, New-Jersey, Pennsylvania, the three lower counties of Newcastle, Kent and Sussex on Delaware, Maryland, Virginia, North-Carolina, and South-Carolina, deputed to represent them in a continental Congress, held in the city of Philadelphia, on the 5th day of September, 1774, avowing our allegiance to his majesty, our affection and regard for our fellow-subjects in Great-Britain and elsewhere, affected with the deepest anxiety, and most alarming apprehensions, at those grievances and distresses, with which his Majesty's American subjects are oppressed; and having taken under our most serious deliberation, the state of the whole continent, find, that the present unhappy situation of our affairs is occasioned by a ruinous system of colony administration, adopted by the British ministry about the year 1763, evidently calculated for enslaving these colonies, and, with them, the British Empire. In prosecution of which system, various acts of parliament have been passed, for raising a revenue in America, for depriving the American subjects, in many instances, of the constitutional trial by jury, exposing their lives to danger, by directing a new and illegal trial beyond the seas, for crimes alleged to have been committed in America...

They use more specific language, too;

North-America... British America... with any colony or province, in North-America

But also say;

as they cannot be received in America; and if any merchant, residing in Great-Britain or Ireland, shall directly or indirectly ship any goods, wares or merchandize, for America, in order to break the said non-importation agreement... to the end, that all such foes to the rights of British-America may be publicly known, and universally contemned as the enemies of American liberty; and thenceforth we respectively will break off all dealings with him or her.

as impose or continue duties on tea, wine, molasses, syrups paneles, coffee, sugar, pimento, indigo, foreign paper, glass, and painters' colours, imported into America, and extend the powers of the admiralty courts beyond their ancient limits, deprive the American subject of trial by jury... by which any persons charged with committing any of the offenses therein described, in America, may be tried in any shire or county within the realm...

So they had already clearly defined themselves as Americans, specifically referring to British North America (since they didn't have any claim to Spanish North America or anywhere else on the two continents outside British colonies signed up in the Association). But, there was no USA yet.

While we cant be certain which draft included it first, both of our earliest documents use the term USA. John Dickinson's Articles of Confederation and Perpetual Union, drafted June 1776;

Art I. THE name of this Confederacy shall be "THE UNITED STATES OF AMERICA"

And Thomas Jefferson's draft of the Declaration of Independence, also from June 1776, declared;

A Declaration of the Representatives of the UNITED STATES OF AMERICA, in General Congress assembled.

The AoC were accepted in Nov of 1777 with a slight wording change to Art I but keeping the name the same. The DoI was officially accepted in finalized form on July 4th, 1776, so it established the name United States of America, officially, in the heading;

IN CONGRESS. JULY 4. 1776.

The Unanimous Declaration of the Thirteen United States of America

When. in the course of human events. ...

The Constitution just continued usage already established by an earlier congress. Interestingly, Joseph Galloway's Plan of Union (1774) uses the term America and would have given us partial autonomy;

That a British and American legislature, for regulating the administration of the general affairs of America, be proposed and established in America, including all the said colonies; within, and under which government, each colony shall retain its present constitution, and powers of regulating and governing its own internal police, in all cases whatsoever.

And the next year (1775) Dr Franklin introduced something more specific. Had we opted for his plan, it would have been a different America;

Art. I. The Name of the Confederacy shall henceforth be The United Colonies of North America.

Hope that gets to the root of your question. Happy Memorial Day, everybody.