BERLIN—This year, the 27-nation European Union was supposed to come of age as an actor on the world stage, bolstered by the Lisbon Treaty, which streamlines the EU's cumbersome institutions. Instead, Europe is starting to look like the loser in a new geopolitical order dominated by the U.S. and emerging powers led by China.

When the world's policy and economic elite gather Wednesday in Davos, Switzerland, for the annual World Economic Forum, much of the talk will be about the rise of a "G-2" world where the U.S. and China...