When cracks recently appeared in beams of the European Parliament's main chamber, forcing its closing, one member, Nigel Farage of the U.K. Independence Party, proclaimed that he would "work for the day that the whole democratic facade of the European Parliament is shut as well."

Mr. Farage is an avowed anti-European known for extreme views. But even for Europeans who do not actively resent the Parliament, it has become a powerful symbol of how institutions designed to build a united Europe have faltered as the project faces the most serious crisis of its 60-year history.