The video of the May 1 event shows Mr Benalla dragging off a woman then beating a young man on the ground.
The Paris prosecutor, which was unaware of the matter before Thursday, launched a preliminary investigation against Benalla on suspicion of violence, usurping the function of a police officer and using signs reserved for public authorities.
Supporters of Macron claimed that the punishment handed down to Benalla - suspension without pay for two weeks and a transfer to an administrative job - was appropriate.
The man is heard begging him to stop, with officers at the scene not intervening.
"The staff member, Alexandre Benalla, had been given permission to witness the demonstrations only as an observer", Mr Roger-Petit said.
Condemning the "unacceptable behaviour", Mr Macron's spokesman Bruno Roger-Petit said that Mr Benalla was removed from organising security for presidential trips.
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Opposition MPs suggested Thursday that there had been a cover-up for Benalla and questioned why the incident on the picturesque Rue Mouffetard, several kilometres from the main demonstration, had not been referred to the police when it came to light.
Given that Benalla's suspension is already complete, it's unclear if his actions would have been revealed to the public if not for the work of French newspaper Le Monde, which identified Benalla earlier this week. "This comes as a punishment for unacceptable behavior", he said.
"I'm surprise he hasn't resigned", said conservative lawmaker Jean-Christophe Lagarde, adding that if Benalla doesn't do so himself, the president should remove him or the drama will jump to "an affair of state".
"The feeling is that at the Elysee people think they're above everything", Ms Wauquiez said.
Later the speaker of the lower house of parliament, Francois de Rugy, announced a parliamentary inquiry into the case.
The 40-year-old centrist was in Australia on May 1, a traditional day of demonstrations in France organised by trade unions, but which was marred this year by hundreds of black-clad anarchists who clashed with police and smashed up shops.