Both are facing serious economic problems. Both will come out of the present situation although with wounds and pain. Europe is fighting to keep its economic union with stronger economies "helping" the weaker members of the Union. Most probably the 27 will maintain their links and uniform currency. New rules and constraints will be imposed to avoid the "big spenders" to get carried away again.
The US will come out at the last minute with an agreement between the two main political parties. However the relevant point here is the size of the fiscal deficit. Undoubtedly the US will have to learn to adjust to higher constraints. Spending has to be reduced and it will be necessary to agree what to cut. Social expense does not look like the accounts to touch momentarily