Out of all the major airlines in the US which is the flag carrier.(Don’t even think about saying Southwest Dami 
)
There’s really no official flag carrier for the USA.
The nearest airline to a flag carrier the USA had, in my opinion, was the original Pan American. For a long time, Juan Trippe’s airline had the only international (outside of Canada) flights. The brought the flag to many intercontinental destinations.
Like dami said, we don’t really have a “flag carrier”. The closest thing at the moment might be American Airlines, but its not official.