Instead, the decision — forced through by a majority vote, over the bitter objections of four eastern members — did as much to underline the bloc’s widening divisions, even over a modest step that barely addresses the crisis.
Nearly half a million migrants and refugees have arrived in Europe this year, according to the United Nations High Commissioner for Refugees, a number that is only expected to rise.
The crisis has tested the limits of Europe’s ability to forge consensus on one of the most divisive issues to confront the union since the fall of Communism. It has set right-wing nationalist and populist politicians against Pan-European humanitarians, who have portrayed the crisis in stark moral terms.
“We would have preferred to have adoption by consensus, but we did not manage to achieve that,” Jean Asselborn, the foreign minister of Luxembourg, said after a meeting of home affairs and interior ministers.
Leaders from across the 28-member bloc will meet in Brussels on Wednesday for further discussions on how to respond to the crisis.
Mr. Asselborn said even countries that voted against the distribution of asylum seekers — the Czech Republic, Hungary, Romania and Slovakia — must comply. “I have no doubt they will implement these decisions fully,” he said.
But with the prime minister of Slovakia immediately threatening to defy the plan, the outcome was more than an example of the bloc’s inability to coordinate its policies — formidable enough through the long crisis over the euro and Greece’s debt.
The response to the refugee crisis so far has also raised profound questions about a failure of European principles, a trembling of the pillars on which the bloc was founded more than 20 years ago.
The European Union’s reputation, and its faith in Brussels, have suffered in the past few months, with sharp and vocal divisions among member states and continuing doubts about Greek economic sustainability.
The migrant crisis “risks bursting the E.U. at its weak seams,” said Stefano Stefanini, a former senior Italian ambassador now based in Brussels. “It’s more dangerous than the Greek drama and more serious than the euro, because it challenges fundamental European accomplishments and beliefs.”
With Tuesday’s vote, he said, “the cleavages only get deeper.”
In practical terms, those achievements are most manifest in the bloc’s single currency and the freedom of movement within the borderless, passport-free zone known as the Schengen area. Both are being put to the test as never before.
As with the euro, borderless travel was pushed ahead by the European Union as an essentially political idea, without Brussels having created the rules and institutions capable of coherently maintaining and enforcing it.
And as with the euro, the chaos over the refugees has raised questions about not only how the European Union functions but what it stands for, not least its aspirations to balance justice and security.
“People want to see both compassion and competence from the E.U., and those two things should not be at odds,” said David Miliband, a former British foreign minister and now director of the International Rescue Committee, a nongovernmental agency that helps refugees. “If the E.U. is incompetent, compassion is not enough.”
Formed as a peaceful, humane response to the blood bath of World War II, the bloc has always prided itself on its commitment to decency, including a traditional welcome among member states to accepting refugees.
But the surge of migrants has shredded that welcome and challenged the principles — and many of the practical benefits — of the Schengen zone. Austria, Germany, Hungary and Slovenia have all re-established border controls, at least temporarily, in recent weeks.
There is a growing recognition that Schengen can function only if its outermost borders are secure — as well as mounting evidence that they are not. Donald Tusk, the president of the European Council and a former prime minister of Poland, wrote European Union leaders last weekend that “we as Europeans are currently not able to manage our common external borders.”
Piled atop existing European concerns about low growth, high unemployment and high deficits, the migrant crisis is only more fodder for growing nationalist, populist and anti-European Union parties in countries like Britain, Denmark, France and Sweden.
“The European norm that is challenged is the idea of one for all, all for one, under the rubric of solidarity,” Mr. Miliband said. “There is a significant question about whether Europe pulls together in the face of a fundamental challenge, or it cleaves apart.”
A Europe “defined by a beggar-my-neighbor race to the bottom was precisely what the E.U. was created to prevent,” he said.
Volker Stanzel, a former German ambassador and senior official, said the migrant crisis was “a fundamental challenge, but not an existential one.” At the moment, “the internal fighting is ever more heated by the day,” he said.
“We’re in a process that is ugly, that some people call ‘refugee poker,’ with everyone horse trading and fighting for their own skin, but doing so in the framework of existing European mechanisms,” Mr. Stanzel said.
That is true even in Germany, Europe’s most European-minded power, he said.