Can I say "many of the United States"?
Solution 1:
I am not sure whether many of the United States would actually be incorrect, but it would certainly be unusual, and I expect it would take your readers a moment or two to figure out what you meant.
I would suggest many U.S. states instead.
Solution 2:
I would suggest using many states within the USA.
[Coincidentally, just before reading this question, I had had to address the same question elsewhere, and that is the expression I used.]