BRUSSELS — When Ursula von der Leyen, the incoming chief of the European Union’s powerful administrative branch, unveiled her top team at a news conference on Tuesday, one new job title provoked a sea of puzzled faces: “vice president for protecting our European way of life.”

But when she then tried to explain what the role would involve — responsibility for migration policy, among other things — the puzzlement turned to outrage. Critics denounced the phrasing as an echo of far-right rhetoric that identifies Europe as white and Christian, and migration from the Middle East and Africa as a threat to that identity.

The outrage has only grown, with pressure mounting on Ms. von der Leyen to scrap the title altogether. By Thursday, even the man Ms. von der Leyen is scheduled to succeed, Jean-Claude Juncker, the current president of the European Commission, was voicing doubts.

“I don’t like the idea that the European way of life is opposed to migration. Accepting those that come from far away is part of the European way of life,” Mr. Juncker told Euronews in an interview.