As hundreds of demonstrators made their way through Lower Manhattan on Tuesday to protest the detention of a prominent pro-Palestinian activist at Columbia University, the White House defended the arrest and rebuked the school for what it called lack of cooperation.

The activist, Mahmoud Khalil, was a leader of student protests on Columbia’s campus and often served as a negotiator and spokesman. Mr. Khalil, 30, who is Palestinian and was born and raised in Syria, is a legal permanent resident of the United States and is married to an American citizen.

He was arrested on Saturday and transferred to detention in Louisiana.

A spokeswoman for President Trump, Karoline Leavitt, told reporters on Tuesday that the government had the authority to revoke Mr. Khalil’s green card under the Immigration and Nationality Act.

“This is an individual who organized group protests that not only disrupted college campus classes and harassed Jewish American students and made them feel unsafe on their own college campus, but also distributed pro-Hamas propaganda fliers with the logo of Hamas,” she said.

Her remarks came a day after Mr. Trump vowed that the apprehension of Mr. Khalil was “the first arrest of many to come.”

Some free speech groups and civil rights activists have questioned the legality of Mr. Khalil’s detention, which his lawyers have challenged in court. On Tuesday, some New York Democrats expressed concern about the arrest. But Mayor Eric Adams shrugged off questions about it at a City Hall news conference, saying that the federal government, not the city, had authority over the matter.

It is rare for the government to move to deport a permanent resident, especially when the resident has not been convicted or accused of a crime. On Monday, a federal judge in Manhattan ordered the government not to deport Mr. Khalil while his case is pending.

Ms. Leavitt, in her comments Tuesday, said that Columbia had been provided with the names of other people she said had “engaged in pro-Hamas activity,” but that it had not helped the Department of Homeland Security identify them.

On Monday, Columbia’s interim president, Katrina Armstrong, sent a letter to the campus saying that the university would follow the law, and that the school’s administration had not requested that federal immigration agents come to the school. A spokeswoman for Columbia made no further comment Tuesday in response to Ms. Leavitt’s remarks.

In New York, Mr. Adams and Gov. Kathy Hochul shied away from condemning or endorsing the arrest. Mr. Adams said that the case was strictly a federal affair.

“I cannot say this clear enough,” the mayor said. “Federal government deals with immigration.”

Mr. Adams has largely refrained from publicly criticizing Mr. Trump, whose Justice Department is seeking the dismissal of Mr. Adams’s corruption case.

He chided reporters for asking repeatedly about the arrest.

“What I’m finding surprising is the level of support you’re all displaying, but I didn’t see that support for me,” he said, alluding to his five-count criminal indictment.

A year ago, Mr. Adams sent police officers in riot gear onto the Columbia campus and arrested more than 100 pro-Palestinian demonstrators.

In Albany, Ms. Hochul told reporters that she did not have enough information to assess whether the arrest of Mr. Khalil was appropriate. She said that immigration enforcement should focus on people here who break the law.

“I just do not have all the facts,” she said. “But here in New York, unlike the president, we actually respect rule of law. So I’m waiting to see what the judge says about this.”

Other New York Democrats were more vocal. State Senator Andrea Stewart-Cousins, the Senate majority leader, called the arrest “chilling” and said that it should be “concerning to all of us if all of a sudden, you can be swept away to another state.”

Senator Chuck Schumer of New York wrote in a social media post Tuesday that, while he may “abhor many of the opinions and policies that Mahmoud Khalil holds and supports,” the federal government should be transparent about whether he is being charged with a crime.

“If the administration cannot prove he has violated any criminal law to justify taking this severe action and is doing it for the opinions he has expressed, then that is wrong,” he wrote.

Hundreds of people turned out Tuesday afternoon to protest the arrest, gathering first in Washington Square Park and then marching downtown to City Hall.

Banners and signs read “Free Khalil,” and marchers chanted “No ICE, no KKK, no fascist USA.” Police helicopters circled.

Police officers detained about a dozen demonstrators about 4:30 p.m. One demonstrator yelled out “Free, free Palestine” while being loaded into a police vehicle.

One of the protesters, Nathan Gibbs, said he believed the arrest of Mr. Khalil posed a threat to the First Amendment.

“It was an incendiary act,” said Mr. Gibbs, a fitness instructor who graduated from Columbia in 2017.

Reporting was contributed by Nate Schweber, Anvee Bhutani, Kaja Andric, Sharla Steinman, Dana Rubinstein, Benjamin Oreskes and Emma G. Fitzsimmons.