The incident took place after a woman was pepper-sprayed by a police officer, eyewitness Vidhushan Vikneswaran told Reuters after filiming the incident.
In a separate incident, French police fired tear gas to disperse protesters from the Champs Elysees as a few hundred protesters from the grassroots 'yellow vests' movement tried to occupy it.
President Emmanuel Macron had presided over the Bastille Day military parade alongside other European leaders.
The famous boulevard was reopened to traffic as soon as the parade finished, but a few hundred protesters tried to occupy it.
Macron and his guests had already left for the Elysee presidential palace for lunch.
The Police Prefecture said on Twitter it had ordered the protesters to leave the area, or be forcibly removed.