*Written in 1787, ratified in 1788, and in operation since 1789
You can legitimately say that the government that exists now can only be traced back to the Constitution, I suppose. But for the Country Known As The United States of America, there's the clear statement in the Declaration:
the Representatives of the united States of America, in General Congress, Assembled, appealing to the Supreme Judge of the world for the rectitude of our intentions, do, in the Name, and by Authority of the good People of these Colonies, solemnly publish and declare, That these United Colonies are, and of Right ought to be Free and Independent States.
That's the official notice that they were united, going to be States instead of colonies. However, for them officially calling themselves the United States of America, that first happens with the Articles of Confederation, in 1777;
Article I. "The Stile of this confederacy shall be, “The United States of America.”
If this was about trademarks, brand names, Doing Business As , and copyrights, I'd go with the Articles.