My instinct was to choose U.S.’s, but I consulted my usual sources to be sure. Unfortunately, I couldn’t find any guidance on how to form the possessive of an abbreviation that ends with an s followed by a period. (While both U.S. and US are acceptable abbreviations, U.S. is more common in the United States and US in the rest of the world.)
What I did find, though, is that when a proper noun is used as an adjective, it’s not a possessive, and therefore doesn’t require an apostrophe. So, for example, it would be proper to write U.S. interests rather than U.S.’s interests. My recommendation is to follow this usage and avoid the problem of the possessive altogether. For instance, instead of writing the U.S.’s first president, you could use the first U.S. president. It looks much cleaner.
If United States is spelled out, the possessive is formed with an apostrophe but no s (for example, The United States’ first president was George Washington.) Since United States is plural in formation, it’s treated as a plural noun, even though it’s singular in usage.
I submitted a question to the Chicago Style Q&A to see what they recommend for the possessive of U.S. If they answer it, I’ll update this post with their reply.