Within the Star Trek universe, the USS doesn't stand for United States Ship, it stands for United Star Ship (or United Space Ship in the earliest episodes).
TRELANE: You must excuse my whimsical way of fetching you here, but when I saw you passing by I simply could not resist.
KIRK: I'm Captain James Kirk of the United Star Ship Enterprise.
The Squire of Gothos
OLD MAN: They're men. They're humans.
PIKE: Captain Christopher Pike, United Space Ship Enterprise.
It's also named as the United Star Ship (the S.S. Yorktown) in the original studio pitch by Gene Roddenberry
You are therefore posted, effective immediately, to command the following:
The S.S Yorktown.
Cruiser Class — Gross 190,000 tons
Crew Complement — 203 persons
Drive — space-warp. (maximum velocity .73 of one light-year per hour)
Range — 15 years at galaxy patrol speeds
Registry -- Earth, United Space Ship
In answer to your supplementary questions, no, the United States no longer exists in the 22nd Century. It's part of a single global government known as United Earth.