Ibrahim Hooper, a spokesman for the Council on American-Islamic Relations, said the response of Muslim-Americans on Sept. 11 was disgust.

“I know because I wrote it,” he said of the council’s reaction, adding that if Mr. Trump had evidence of cheering, he should present it.

Hope Hicks, a spokeswoman for Mr. Trump, would not elaborate on his comments; his aides have repeatedly declined to make him available to address the controversies over some of his statements. Ms. Hicks said only that Mr. Trump had drawn an “unprecedented” crowd of 10,000 in Birmingham, adding, “Mr. Trump’s speech was great and unbelievably well received.”

Since the Nov. 13 attacks in Paris by Islamic extremists, which killed 130 people, the national mood has turned against admitting refugees from Syria, where the Islamic State has taken control of certain areas, a poll by Bloomberg Politics showed last week. And Mr. Trump has been riding a wave of anger and mistrust of government and politicians since he entered the race in June.

Mr. Trump’s calls for surveillance came after he said earlier in the week that he supported closing some mosques. He has not been alone in making such calls; a top donor to Hillary Rodham Clinton also urged such surveillance in the wake of the Paris attacks. The donor, Haim Saban, later said he had misspoken.

The rest of Saturday’s speech was a classic Trump performance, a crowd-pleasing stream of boasts, humor and mockery. Referring to George E. Pataki, the former New York governor who has been lagging badly in polls and was one of several opponents to seek airtime from NBC after Mr. Trump hosted the network’s “Saturday Night Live,” he said: “They could give him 12 ‘Saturday Night Lives,’ an hour and a half a night — not going to have any impact.”