The chain of events began in January, when David Streever sent an email to the acting director of the Immigration and Customs Enforcement agency after two demonstrators had been shot to death in Minneapolis. Subject line: “What’s next.”

The email called the agency’s acting director a “monstrous human being,” compared him to a Nazi official and said that he would be tormented by his conscience.

Then, in June, there was a knock at the door of Mr. Streever’s home in Rochester N.Y. Federal agents had come with a written warning that said Mr. Streever may have violated a law against threatening a federal law enforcement officer.

Now, a free-speech watchdog is suing the Trump administration, arguing that the agents’ actions amounted to unconstitutional retaliation for protected speech and were intended to intimidate Mr. Streever into withholding his criticism of federal immigration policy.

The lawsuit, filed Monday in the U.S. District Court for the District of Columbia, names as defendants Homeland Security Secretary Markwayne Mullin; the acting director of ICE, David Venturella; and the agents involved in the investigation into Mr. Streever. It asks the court to declare their conduct a constitutional violation and to prohibit similar investigations.

“Our Constitution does not tolerate such a brazen abuse of authority,” the lawsuit, which was filed by the Foundation for Individual Rights and Expression, says.

The case arises from an episode that has drawn national attention and intensified concerns among civil liberties advocates about the government’s response to criticism.

Mr. Streever, 45, was on a trip to Finland with his 7-year-old daughter on June 23 when two investigators from the Department of Homeland Security visited his home and presented his wife with a document for her husband that read “Warning notice” and “You may be in violation of federal law.”

It said that the email Mr. Streever had sent to Todd Lyons, who was the acting ICE director at the time, may have constituted a threat.

Two days later, a third agent attempted to intercept Mr. Streever at a New York City hotel where he and his daughter were staying after returning to the United States, according to the lawsuit.

“When you have federal agents showing up at your home, confronting your wife, tracking you to a hotel with your daughter, that’s more than enough to send the message that you should shut up,” said Adam Steinbaugh, a lawyer at the Foundation for Individual Rights and Expression.

The Department of Homeland Security, the parent agency of ICE, issued a statement in response to questions about its warning notices and the lawsuit, and said that the agency “investigates all credible threats toward its employees and officers, including threats to the ICE director.”

“As a matter of policy, we do not comment on any ongoing investigations,” the statement said.

Mr. Streever, a former journalist who works in the tech industry, sent his email to Mr. Lyons after the immigration crackdown in Minneapolis and the fatal shootings of two demonstrators, Renee Good and Alex Pretti, by federal officers.

The email, which is included in full in the lawsuit filed on Monday, said that Mr. Lyons “will never know peace,” and that he “will go down in history as America’s Reinhard Heydrich, the butcher,” a reference to the Nazi security chief considered to be a principal architect of the Holocaust.

“Even Trump will turn on you before the end, and you will be a sad, despised man who eats himself alive with shame at your own pathetic weakness,” the email read.

“You will seek to lose yourself, to escape the burden of knowing the truth about yourself. But wherever you go, you will find yourself. You will torment yourself until your last day on Earth.”

Mr. Steinbaugh cast the email as “a moral condemnation.”

“This is pleading with a government official to listen to his conscience and warning that his conscience is not going to give him peace in the future,” he said. “That’s a rhetorical appeal. That’s not a threat.”

The warning letter from the Department of Homeland Security invited Mr. Streever to contact the agency, but he has no intention of doing so, his lawyer said.

Nathan Freed Wessler, the deputy director of the American Civil Liberties Union’s Speech, Privacy and Technology Project, criticized the government’s warning notice as an intimidation tactic.

“Sending a frustrated or even intemperate email to a federal official is at the core of what the First Amendment protects,” he said.

The lawsuit references what it cites as other recent instances of the immigration agency seeking to suppress protected speech.

One involved the same agents who visited Mr. Streever’s home issuing a similar warning the same day to a poll worker in Syracuse, about an hour’s drive from Rochester.

Paigelynne Gonyea was working at a voting location in a library during New York’s primary elections when the agents arrived.

Ms. Gonyea, 40, said that she believed her warning stemmed from a post she had made on social media in January in which she posted a photo of Jonathan Ross, the ICE officer who shot Ms. Good, and wrote, “I think today is a great day for Jonathan to be indicted!”

A Department of Homeland Security spokeswoman, however, shared an image of a different social media post with The Associated Press that the agency attributed to Ms. Gonyea and that contained Mr. Ross’s home address. The spokeswoman said the post was evidence of “doxxing,” the posting of personal information online without consent.

The interactions between the agents and Mr. Streever and Ms. Gonyea were first reported by Syracuse.com.

Ms. Gonyea declined to discuss her social media posts when reached by The New York Times and said that she was pursuing legal options.

“I think this was done as an intimidation tactic,” she said.