The incident has since become the focus of a political dispute over how it is described.
The officer was taken to hospital.
Police said the man was overpowered and detained, and that a knife was found hidden under his jacket.
Private broadcaster TVN24 reported that he is set to answer for assaulting an officer.
According to police, officers securing the demonstration noticed the man behaving nervously and moved to check his identity.
He ran into a building on Marszałkowska Street and became aggressive inside.
TVN24, citing unofficial information, reported that he pushed the officer into a glass door, which shattered, and that he briefly lost consciousness during the arrest.
Right-wing media had described the incident as an attack on the office of Adam Borowski, who opposed Poland’s communist-era government and has been linked in recent years to the Law and Justice (PiS) party.
They reported that the aggressor was a woman and that a man connected to the protest was beaten.
TVN24 said the incident took place at a building housing the Free Speech Association, where Borowski sits on the board.
PiS leader Jarosław Kaczyński wrote on X about a "forceful attack."
In a statement, police said they were responding to public claims that blamed officers for the incident.
They said those on duty should not be held responsible for aggression against them, and that many of the reports circulating were disinformation.
Separately, Interior Minister Marcin Kierwiński accused the conservative TV Republika channel, which was broadcasting the protest live, of vulgarly" insulting police.
A news ticker on the channel read "This is not police, it’s pathology," and a protester called officers "trained dogs."
Kierwiński said anyone who attacks Polish police would face consequences.
Protesters opposed to changes at the Constitutional Tribunal have gathered outside its headquarters for several weeks.
Tensions flared outside the court on Wednesday after Maciej Berek was sworn in as a judge in parliament, with protesters clashing with police.
The Tribunal's president Bogdan Święczkowski has not allowed him to rule on cases.
(ał)
Source: PAP, tvn24.pl, wp.pl