European Commission President José Manuel Barroso brimmed with optimism as the first decade of the 21st century came to a close. The European Union, he asserted, is “uniquely suited to take on its leadership responsibilities” for a world that undergoes rapid change.

Instead, the very survival of the European project has seemed in doubt at times in the ensuing years. An intensifying rivalry between the U.S., a rising China and an assertive Russia has reshaped the world order, undermining a rules-based international system...