The USA existed before the constitution was written and was governed under the Articles of Confederation which were adopted in 1787. The Constitution merely changed the structure of the government it did not create a non existent country, and the relationship of the states to each other. The declaration of Independence also styled the colonies as the united States of America. The Articles of Confederation also make it explicit that the US was the US before the constitution was written as can be seen below in Article I.
The Articles of Confederation
Agreed to by Congress November 15, 1777; ratified and in force, March 1, 1781.
Preamble
To all to whom these Presents shall come, we the undersigned Delegates of the States affixed to our Names send greeting.
Articles of Confederation and perpetual Union between the States of New Hampshire, Massachusetts bay, Rhode Island and Providence Plantations, Connecticut, New York, New Jersey, Pennsylvania, Delaware, Maryland, Virginia, North Carolina, South Carolina and Georgia.
Article I. The Stile of this Confederacy shall be "The United States of America."
Did it ever! The Federal Constitution changed everything. They still called it the United States of America. America remained underneath, untouched where it still is no matter how many layers of state are piled on top.
That's a bit like saying "The invention of the car really just changed the structure of the horse and buggy."