I don't think there's exactly a canon answer to this, but I was curious based on the way Pike and Spock discuss the United States in the first episode. The way Pike asks Spock if he's aware of the United States implies it existed in the past tense. And one would think that if it still existed, the question would be silly, since Spock and everyone would know what the United States is based on the fact Starfleet is headquartered in San Francisco.
Enterprise implies the United Kingdom still exists within United Earth, since Malcolm Reed's family is said to have wanted him to serve in the Royal Navy. So it would be possible for the US to exist with United Earth too. We also know that the individual states still exist since Kirk is from Iowa and Riker is from Alaska.
The furthest out that Trek has stated the US will still be around is 2079 (i.e., the same year as Q's court of post-apocalyptic horrors), since TNG's "The Royale" shows a NASA space craft whose occupant is wearing a uniform with a US flag that has 52 stars, with Data stating the flag represents a time period from 2033 to 2079.
Enterprise implies the United Kingdom still exists within United Earth, since Malcolm Reed's family is said to have wanted him to serve in the Royal Navy. So it would be possible for the US to exist with United Earth too. We also know that the individual states still exist since Kirk is from Iowa and Riker is from Alaska.
The furthest out that Trek has stated the US will still be around is 2079 (i.e., the same year as Q's court of post-apocalyptic horrors), since TNG's "The Royale" shows a NASA space craft whose occupant is wearing a uniform with a US flag that has 52 stars, with Data stating the flag represents a time period from 2033 to 2079.