While US states saw similar affects on their budgets (since they are not allowed to run deficits) this was offset by large federal deficits which kept the economy running. Borrowing costs were kept low by the Fed's QE.
EU central spending is only about 1% of GDP, much less smaller than the US Federal Government spending of 20% in normal years and does not borrow in its own right. The ECB's eventual agreement to stand behind Eurozone governments and pledging to buy their debt was crucial to resolving the crisis.