Macron has also added an global dimension to his anti-extremism efforts, asking Russian counterpart Vladimir Putin for strengthened cooperation in the fight against terrorism during a phone call on Tuesday.
On Tuesday, Mr Macron said the Sheikh Yassin Collective - an Islamist group named after the founder of the Palestinian militant group Hamas - would be outlawed for being "directly involved" in the killing.
Both men were detained by police after Paty's killing.
Macron spoke following a meeting with anti-terrorism police in the Paris suburb of Bobigny. "These actions will be stepped up".
The French minister's visit was cut short, however, due to the Paris attack.
"One of our compatriots was murdered today because he taught...the freedom of expression, the freedom to believe or not believe", Macron said.
The suspect, identified as 18-year-old Aboulakh A, was a Moscow-born Russian national of Chechen origin who had been granted a 10-year residency as a refugee in March 2020, according to The Sun.
In the video, the father had reportedly called Paty a "rogue" who should no longer be a teacher and called on other parents to mobilise.
The father of a 13-year-old girl in Paty's class, who may not have been in school the day of the lesson, published a series of Facebook videos denouncing the lesson and calling for support in protest.
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Anzorov was heard to shout "Allahu Akbar" - the Islamic phrase God is the Greatest - before posting a video of the teacher's decapitated head on social media.
Nine people, including members of Anzorov's family, were placed in custody in the early hours of yesterday.
France's Minister of the Interior, Gerald Darmanin, has asked French security services to deport 231 irregular migrants, including nine Moroccans, due to their "extremist beliefs".
Education Minister Jean-Michel Blanquer said Tuesday that Paty had been "martyred" because of his profession, and would be posthumously bestowed France's highest order of merit, the Legion of Honor.
Witnesses said the suspect was spotted at the school on Friday asking pupils where he could find Paty. The magazine had similarly published caricatures of the Prophet Muhammed.
Paty's beheading was the second knife attack claimed in the name of avenging the Prophet since a trial started last month over the Charlie Hebdo killings in 2015 when 12 people, including cartoonists, were gunned down for publishing Mohammad cartoons.
At a square in eastern Paris, people offered flowers to the male victim, and held up signs declaring that they, too, are teachers.
Tens of thousands of people rallied in Paris and cities across France in solidarity with a teacher beheaded.