The United Auto Workers union suffered a crushing defeat Friday, falling short in an election in which it seemed to have a clear path to organizing workers at Volkswagen AG's plant in Chattanooga, Tenn.

The setback is a bitter defeat because the union had the cooperation of Volkswagen management and the aid of Germany's powerful IG Metall union, yet it failed to win a majority among the plants 1,550 hourly workers.

Volkswagen...