White people aren't suddenly disappearing, they just aren't a majority any longer (perhaps because people of hispanic and/or mixed ethnicity aren't counted as white?)

They will remain a sizeable fraction of the population for the foreseeable future, if not forever, just like most other ethnicities in the US. I don't expect white Americans becoming 30-40% of the population instead of 50% makes much of a difference.

Another thing to keep in mind: there is no single group that runs the USA even today. The voter base is comprised of American citizens, a large fraction of whom aren't white. Simply think back to how often you've heard about the "black vote" being key to winning one state or another, and consider that every demographic may tend to vote for certain policies over others.

I honestly think that in the future, ethnic identity will play way less of a role in the political landscape of the US, which I can only see as a good thing, bringing about a more equitable environment for all Americans.