Or it could be split between the remnants of the North and the South, both claiming to be the United States like one user as suggested. Plus, European immigration is going to become very common in Canada (Ireland is part of Britain), Mexico (The strong amounts of Catholicism), and Louisiana (New Orleans), especially with Catholics.
I like the idea of two states both claiming to be the true United States. There should also be "true names" for each of them that the other uses to de-legitimize the other one. Like the southern "America" calling the northern "America" the Federals again or "Ohio". Or the northern "America" calling the southern America "Dixie".
With immigration avoiding America like the plague I could see both Americas being extremely underpopulated. Even after at least one of the American successors manages to build up a decent industry.
Part II of my last comment:
I still think that Canada could call themselves a Kingdom with no powerful America to (hypothetically) raise a fuss over it.
I just realized that northern Mexico is going to be an interesting melting pot. California's going to be a hodge-podge of cultures and I'm thinking that there could religious refugees if President-For-Life Calhoun decides that the separation of church and state is another thing not necessary in his new America.