That bears repeating. What will happen is twofold - first, that to the EU will go the labor that that sort of society needs - low-paying, manual "dirtywork" labor that forms the lower strata of an increasingly class-based, aging welfare society. And second, that to the United States will go the young, highly-educated, risk-tolerant innovators who are looking for the rewards success brings there but not in Europe, and worse from the EU point of view, these won't just be from the third-world countries populating the EU but from the EU's best and brightest as well.
This is not the recipe for pulling even with the United States in terms of technology, culture, or economics. But it is an old pattern.
The problem with that is that the old European welfare state is on the verge of total and utter collapse, probably within the next five to ten years or so (our American welfare state is in danger in some ways itself, but the situation is not nearly as dire here). What will take place in Old Europe once that happens is uncertain at this point, but I guarantee it won't be pretty. We're on the cusp of some very interesting, and possibly frighteningly violent, times ahead of us.