I see this claim made quite often.
I have never seen any actual evidence to support the claim. There are documents and speeches in which “these” is used, rather than “the,” but there are plenty of references using “the.”
Jefferson’s request for funds for the Lewis & Clark expedition (1803) uses “the United States” and “the US.”
Marshall’s opinions in Marbury v Madison (1803) and McCulloch v. Maryland (1819) repeatedly use “the United States” without ever referring to “these United States.”
The Missouri Compromise (1820) repeatedly uses “the United States” without once using “these.”
Monroe Doctrine (1823): every reference is to “the United States.”
Andrew Jackson’s congressional message “On Indian Removal” (1830): every reference is to “the United States.” . (We cannot even get Jackson to use “these” instead of “the”?)
The Treaty of Guadalupe Hidalgo (1848, ending our assault upon Mexico): every reference is to “the United States.”
Compromise of 1850: every reference is to “the United States.”
Kansas-Nebraska Act (1854): every reference is to “the United Staes.”
Dred Scott v. Sanford (1857): every reference is to “the United States.”
The Homestead Act (1862): every reference is to “the United States.”
This is not to say that the word “these” was never used as a rhetorical flourish to make a point in one essay or speech or another. However, the claim does not appear to ever make it into an actual government document. I suspect that dividing “these” from “the” was an invention of Shelby Foote, but I have never seen his evidence, either. (I ascribe no malice to Foote; I simply think he found some “these” references and drew an inaccurate conclusion.)
Among speeches, I found one from Frederick Douglass from 1852 in which he uses “these United States.” Although George Washington’s First Inaugural Address uses “the United States.”