Prosecutors there have opened an inquiry to try to establish what happened Saturday when 73-year-old Genevieve Legay suffered head injuries.
When paramedics evacuated veteran activist Legay from the demonstration she was bleeding from the head, but still conscious.
Her daughter later said that she had suffered several fractures to the skull.
Arie Alimi, the family's lawyer said they would be filing a formal complaint against the authorities for violence against "a vulnerable person".
"The police charge was very violent," he said. "Mrs Legay ... has been very badly injured." But contrary to some reports, she was in a stable condition and not in a coma, he added.
Photos and video footage from Saturday's protest showed her carrying a rainbow-coloured flag with the word "peace" written on it.
Legay is an activist with the social justice campaign group Attac. The group published a message on its website calling for an inquiry to establish who was responsible for her injuries.
In an interview with Nice-Matin published Monday, President Emmanuel Macron wished her "a speedy recovery, and perhaps some form of wisdom".
"When you are frail, and may be pushed around, you do not go to places that are defined as prohibited and do not put yourself in situations like this," he said.
Both local prosecutors and France's justice minister Sunday pointed out that protests in some parts of the city centre had been banned.
The minister. Nicole Belloubet, was asked about the incident in an interview with French channel BFMTV.
While wishing Legay a speedy recovery, she said, "I do find it curious all the same that when a demonstration had been banned, as was the case in Nice, someone goes with the declared intention of demonstrating in that place there."
"There were some areas, in some cities, where demonstrations were banned. Following warnings, a person who stays there is likely to commit a crime and it is in this context that the events happened."
Nice prosecutor Jean-Michel Pretre said they were doing everything they could to find out what happened.