Under growing pressure, Mr Macron gathered several ministers together on Sunday to discuss the row.
Alexandre Benalla, who was Mr Macron's top bodyguard, is seen dragging away a woman and then beating a man during May Day protests in Paris.
He has been charged with group violence and illegally wearing a police badge.
Mr Benalla was fired on Friday.
An official said Mr Macron had described the incident "unacceptable" and promised there would be "no impunity".
Senior presidential official Alexis Kohler is to look into reorganising Mr Macron's private office so as to prevent a repeat of the incident, unnamed officials say.
The French presidency has been accused of being aware of the incident for some time, trying to cover it up, and failing to act swiftly against Mr Benalla.
Three policemen have also been charged in connection with the incident. They were questioned on Saturday for allegedly leaking security footage to try and prove Mr Benalla's innocence.
Vincent Crase, an employee of Mr Macron's La République en Marche (Republic on the Move) party, is also being investigated after he appeared in the video.
Public outrage has been stoked by additional footage that appears to show several police officers watching the incident without intervening.
Interior Minister Gérard Collomb is expected to be questioned on the issue in parliament on Monday.
How did we get here?
The video was posted on social media in May, but the case became a political scandal after Le Monde newspaper revealed that the attacker was Mr Benalla, aged 26.
A former bodyguard of Mr Macron, he was hired as an aide to the president's chief of staff after last year's election.
He was then given an apartment in an upmarket Paris district and a chauffeur-driven car, French media say.
He also had the highest security clearance to parliament.
In May, a few days after the incident, he was suspended for two weeks but nothing was reported to prosecutors.
The BBC's Hugh Schofield in Paris says this suggests that Mr Macron's office may have already been aware of his actions.
What happened on May Day?
The incident took place in a popular tourist spot in Paris' Latin Quarter where about 100 people had gathered.
The original video shows a man wearing a police helmet, but no uniform, joining CRS riot police after clashes erupted.
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