What the US has, and the EU would need, are some mechanisms to level economic differences between the constituent states. Any federal US program effectively works as such, even though that is rarely acknowledged in the public discourse. As but one example: military spending is financed primarily by the economically strong and populous states, and disproportionately favors poorer states with defense contacts or military bases.
The EU has a few such mechanisms, some explicit (regional development funds), some less so (agriculture). But the total E.U. budget is rather tiny compared to the economy. Something like a common unemployment insurance would be best, mechanistically. Because it would automatically adjust to changing fortunes at the very local level without requiring constant political intervention and the accompanying bargaining. At present, such a policy is unlikely to find approval because of pervasive remaining mistrust („lazy Greeks taking our money“).