A decade and a half ago, it wasn’t just the more fanciful observers who thought that Europe was ready to challenge the world’s leading economic superpower. The region was outpacing the United States in growth. Postunification Germany was being hailed as an economic giant in its own right. And in the European Union, the tide of integration euphoria, nurtured by the single-market program and plans for a single currency, was running quite high.

The intervening years have been a disappointment. The “old” continent, to be sure, appears rich, socially progressive, and culturally exciting. It has an impressive crop of global companies. But far from catching up with and overtaking the United States, Europe is finding that the economic cracks in the facade may be getting wider. A huge effort is now required not just to realize the full potential of Europe but also to ensure that an aging population doesn’t overwhelm what it has already achieved. The consequences of failure would be relegation to the second economic division and a blow to Europe’s self-esteem.

The inappropriate way the challenge has been framed is part of the problem: many Europeans, for example, believe that merely matching US growth rates is a...