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Senator Ted Cruz of Texas angered American Muslims on Tuesday with a call to “patrol and secure Muslim neighborhoods” in the wake of the terrorist attacks in Brussels.

Arguing that politicians had “tried to deny this enemy exists out of a combination of political correctness and fear,” Mr. Cruz said that Europeans were “seeing what comes of a toxic mix of migrants who have been infiltrated by terrorists and isolated, radical Muslim neighborhoods.”

He continued, in a statement released by both his campaign and his Senate office, “We need to empower law enforcement to patrol and secure Muslim neighborhoods before they become radicalized.”

Senate and campaign aides to Mr. Cruz did not respond to questions seeking clarification on how the proposal might be carried out.

The comments drew immediate rebukes from Muslim groups who were already wary of Mr. Cruz. Last week, he came under fire after announcing a team of national security advisers that included Frank Gaffney Jr., a former Reagan administration official who is perhaps best known for holding extreme views about Islam. (He once wrote an op-ed in The Washington Times suggesting that President Obama is a Muslim.)

In a phone interview on Tuesday, Nihad Awad, the national executive director of the Council on American-Islamic Relations, called Mr. Cruz a “religious zealot” with little regard for civil liberties.

“We do not have the problem that Ted Cruz envisions,” Mr. Awad said of Muslims in the United States. “He wants to bring to our memories checkpoints in the streets. That’s really going to be similar to third-world countries and to what happened in Nazi Germany.”

Mr. Cruz has at times stopped short of some of Donald J. Trump’s most controversial suggestions — opposing, for instance, calls for a national database of Muslims.

Speaking to reporters in Washington on Tuesday, he deflected when asked if he supported Mr. Trump’s proposal to ban Muslims from entering the United States.

“It is time for us to implement serious vetting,” Mr. Cruz said, “and we should not be allowing anyone to come to this country that we cannot vet to make sure that they are not radical Islamic terrorists.”