Demonstrations started out on Friday after the killings in the 10th district, which is home to a large Kurdish community.
The French government has promised to step up security at community centres but Kurdish leaders want the attack to be treated as terrorism.
Police will be sent to protect Kurdish community sites across the country after the deadly attack, Interior Minister Gerald Darmanin said.
The motives of the alleged attacker, a 69-year-old Frenchman who “as things stand” acted alone, were still unknown, he added.
French authorities have been cautious about suggesting a motive, but have pointed to racism.
The Kurdish Democratic Council of France (CDK-F), an umbrella group for Kurds in France which uses the Paris cultural centre as its headquarters, said in a statement it considered the shooting to be a “terror attack”.
Members of the Kurdish centre wept and hugged each other after Friday’s shooting which, for some, revived memories of the murders of three Kurdish activities in January 2013.
“It's starting again. You aren't protecting us. We're being killed!” one person could be heard shouting at police at the scene.
A representative at the centre, Agit Polat said: “Once more failing to protect us … For us, this is a terrorist attack.”
One woman and two men were killed in the attack, and three men were wounded, Paris prosecutor Laure Beccuau said.
“There is no evidence at this stage to link [the suspect] to any extremist ideological movement,” she said.
Ms Beccuau said the question of whether Friday's attack was motivated by racism “will form part of our investigations”.
President Emmanuel Macron said in a tweet that “the Kurds of France” had been “the target of a vile attack in the heart of Paris”.
The three people killed were all reported to be members of the Kurdish community.
The Kurdish community centre, called Centre Ahmet Kaya, is used by a charity that organises concerts and exhibitions, and helps the Kurdish diaspora in the Paris region.
Police used tear gas to disperse Kurdish protesters as they apparently tried to break through a security cordon set up when Mr Darmanin arrived at the scene.
Demonstrators threw objects at police while voicing fury over an attack they felt was deliberate.
Bricks thrown by protesters broke the windows of cars parked in the area and damaged police vehicles.