Prosecutors there have opened an inquiry to try to establish what happened when 73-year-old Genevieve Legay suffered serious head injuries.
When paramedics evacuated veteran activist Legay from the demonstration she was bleeding from the head, but still conscious, AFP journalists at the scene noted.
Her daughter later said that she had suffered several fractures to the skull and subdural haematomas.
Arie Alimi, the family's lawyer, told AFP that they would be filing a formal complaint against the authorities for violence against “a vulnerable person”.
The region's governor would be named in the lawsuit, to be submitted on Monday.
“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, 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.
(Genevieve Legay at the protest on Saturday. Photo: AFP)
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.
“When you stay in a protest after the regulatory (police) warning that people have to disperse, it's a crime.”
In Legay's case however, it was not clear whether she had been in the banned area, or on the edge of it, when police moved against the protesters.
(There were clashes between police and protesters in Nice, which had banned demos in parts of the city centre. Photo: AFP)
Demonstrations this weekend were also banned in certain parts of Paris and in the centres of Toulouse, Bordeaux, Dijon and Rennes.
French courts so far have convicted around 2,000 “yellow vest” protesters of offences since the marches began last November, Belloubet told BFMTV.
“Of the 2,000 convictions that have already taken place, the figure that should be retained is that 40 percent are prison sentences,” she said.
They ranged from one month to three years, according to justice ministry figures.
The other 60 percent of convictions involved community service, suspended sentences or other non-custodial sentences, the minister added.
Nearly 1,800 cases arising from the protests, in Paris and other French cities every Saturday since November 17, were still to be resolved and a total of 8,700 people had been detained, she said.
More than 40,000 people took to the streets across France on Saturday for the 19th consecutive week of anti-government protests, Interior Minister Christophe Castaner said.
The government redeployed soldiers from its Sentinelle anti-terror force to guard public buildings in Paris, freeing up 6,000 police in Paris to tackle any flare-ups of violence.