Brendan Orsinger does not like the D.C. police chief. He has made his views clear on Twitter, writing that “Peter Newsham is a liability to this city” and calling officers “a bunch of violent bullies.”

Newsham, citing tweets he calls “cruel and nasty” — and sometimes inaccurate — blocked his 36-year-old social media antagonist, and at least one other activist on Twitter. That meant they could no longer follow @ChiefNewsham or see the chief’s tweets.

The chief’s stance appeared to attract attention from not only those critics, but also his boss, Mayor Muriel E. Bowser (D). On Friday, after The Washington Post published an article on the issue, the mayor’s office sent an email to officials at city agencies reminding them of a policy prohibiting the blocking or deleting of social media followers.

“Regardless of what comments we receive on the information we put out, we must continue to build community in the digital space,” the email says. “Therefore, if you receive cruel or racy comments on your social media accounts, do not block followers.”

A police spokesman could not immediately say whether Newsham unblocked followers after the note was sent. But the issue goes to the heart of a larger debate — centering on the intersection of social media and transparent government — playing out across the country.

Advocates of open government argue that Newsham, along with other elected and appointed public officials who have weeded out critics on social media, are running afoul of the First Amendment. Hitting Twitter’s “block” key, they argue, is akin to government cracking down on speakers in what has become the nation’s new town square.

The Supreme Court could end up deciding whether the First Amendment applies to public forums run by government officials on the Internet — whether they are used to post feel-good pictures of employees receiving awards, schedules of meetings or discussions of policy.

The American Civil Liberties Union took Maryland’s governor to court over the issue, and another group is suing President Trump. In Virginia, a federal judge ruled that the chairman of the Board of Supervisors in Loudoun County, Va., violated the U.S. Constitution by blocking one critical Facebook follower for 12 hours. That case is being appealed.

Newsham set up his Twitter account in August 2016 and had posted nearly 400 times as of Saturday. He has about 1,300 followers and appears on the page in uniform against a backdrop of marked police vehicles and a picture of himself greeting children. The account is separate from the main police Twitter account, @DCPoliceDept, which has more than 200,000 followers and sends alerts on shootings, robberies, carjackings, missing people, as well as videos of crime suspects who are being sought. No one has been blocked from that account.

Newsham said he started @ChiefNewsham to post “positive information about the police.” Most of his posts show the chief giving awards to officers for good deeds or thanking others for their help. He retweets some items off the main D.C. police Twitter account, and he sometimes comments on events. When a toddler was wounded, Newsham wrote: “no one shoots a 1yr old and gets away w/it on our watch.”

In an interview, Newsham explained his actions: “I have a rule. If someone regularly tweets cruel and nasty things that are fraught with misinformation, I will delete them, or block them.” He said at least two users were posting tweets “that I felt were unnecessary for me to see.”

Newsham, who also is a lawyer, declined to discuss the legal implications. He said that he believes in communicating with residents in a variety of ways and that if he cannot continue using Twitter under his rules, he might close the account.

Orsinger, who describes himself as an activist and tweets as @ToBeSelfEvident, said that “when you create a Twitter account, you create a forum. By excluding people from a public forum you are in sort of murky water.” Orsinger, who participated in the Inauguration Day protests and in rallies against police after the fatal shooting of an unarmed motorcyclist by D.C. police, often uses the hashtag #NeverNewsham.

Orsinger acknowledges that his tweets “are not flattering” to Newsham but said he feels that he and others — including a critic who goes by @AltChiefNewsham — are being punished as retribution for critical speech. The ­@AltChiefNewsham account has called Newsham a liar who covers up “the abuse & violence” of his officers against “black & brown communities.”

Last year, the office of Maryland Gov. Larry Hogan admitted that the governor had blocked as many as 450 people from his Facebook page. Aides to Hogan (R) said they thought a number of critical messages were part of a coordinated attack. Other followers were banned for what the governor’s office described as “hateful or racist” language.

Hogan spokesman Doug Mayer told The Washington Post at the time: “We encourage debate and all manners of political discourse on the governor’s page. But it doesn’t mean we will let an outside group with their own political motivation to hijack the governor’s page.”

Several Maryland residents backed by the ACLU of Maryland sued Hogan in federal court in Baltimore but settled the case with undisclosed terms.

In New York, the Knight First Amendment Institute is suing Trump after he blocked accounts on Twitter. Defending the practice, lawyers at the Justice Department said in court papers that the argument that followers “have been denied access is baseless. At most, the account is a channel for speech.” The attorney wrote that the president “uses the account for his speech, not as a forum for the private speech of others. And his decision to block certain users allows him to choose the information he consumes and the individuals with whom he interacts — expressive choices that public officials retain the right to make.”

The government’s lawyers wrote that Trump, like all users, abides by the rules established by Twitter, not by the government. “The President on Twitter serves as a participant in, not a regulator of, the marketplace of ideas,” the government argued.

Lawyers on all sides are paying close attention to the Court of Appeals for the 4th Circuit, which will be hearing the appeal from Phyllis J. Randall, the chair of the Loudoun County Board of Supervisors. In July, U.S. District Judge James C. Cacheris concluded that the free-speech rules that govern officials in public life follow them to the Internet.

“Social media — and Facebook in particular — has become a vital platform for speech of all kinds,” Cacheris wrote in his ruling. “Indeed, social media may now be ‘the most important’ modern forum ‘for the exchange of views.’ The First Amendment applies to speech on social media with no less force than in other types of forums.”

The judge did note that the case “raises a novel legal question: when is a social media account maintained by a public official considered ‘governmental’ in nature, and thus subject to constitutional constraints?” He noted that Randall used her title when naming the social media page and had “used it as a tool of governance.”

Jameel Jaffer, the director of the Knight First Amendment Institute at Columbia University, which is suing Trump, said public officials are still experimenting with communicating through social media and “are sometimes learning as they go.”

But, he added, “the one thing they need to learn is when they use this medium for official purposes, the Constitution is going to apply . . . . All the protections that come with the First Amendment also apply in digital space.”

This article has been corrected to refer to Brendan Orsinger as an activist.