The child and his mother were crossing the Promenade des Anglais, the famous palm-lined street overlooking the Mediterranean, at a pedestrian crossing when the accident happened on Wednesday.
The boy “was holding a bag that his mother was also holding, but was walking a bit ahead of her”, police said, adding they were “hidden by street furniture”.
A 40-year-old on an electric scooter who was going “at excessive speed” could not avoid the child, they said.
“Despite being injured and having fallen, the rider of the electric scooter immediately cared for the child,” they added.
Nice mayor Christian Estrosi, on Twitter, expressed his “deep emotion” over the incident and opposition to self-service scooters.
“We are studying all possible solutions for avoiding these tragedies,” he said.
According to police in the Riviera city, accidents involving electric scooters are increasingly occurring, especially on roads.
In Paris authorities have brought in strict limits on e-scooters, including speed limits and a limit on the number of operators allowed, after a spate of problems.
The boy’s family had arrived in France at the beginning of Russia’s invasion of Ukraine, the head of a French-Ukrainian association in the Riviera, Iryna Podyriako, told the Nice-Matin regional daily.