Hundreds of masked protesters hurled projectiles at police on police during the demonstration, before turning their attention on buildings, cars, bikes, and street signs as they marched across Paris.
The unrest continued into Tuesday night, most notably as rioters known as casseurs ("breakers") in France vandalized the renowned Necker children's hospital in the 15th arrondissement of southern Paris.
The vandalism was caught on camera as it happened, apparently carried out by two hooded people.
Health Minister Marisol Touraine called the attack "the revolting work of rioters".
"We are talking about a children's hospital that was attacked. There were children going into the operating room at the time, some were asleep. This is truly shocking," she told France Info.
The rioters reportedly smashed 15 windows at the hospital, and social media images from the scene showed that rioters spray painted graffiti on the building.
The Tweeter below points out that the words "Never work" were sprayed on the hospital, adding that she considered it an "absolute indecency".
The authority that runs hospitals in Paris will be pressing charges.
Later in the night, several cars were torched by the rioters, including one in the central République square (see photo below) as the rioters went on the rampage across Paris.
The vehicle belonged to Paris transport body RATP, and had no one inside at the time.
Later in the night, two Autolib rental cars were torched on Rue Saint-Maur in the 11th arrondissement, reported BFM TV.
There were no people inside these cars at the time either, the channel reported.
A total of forty people were hurt and dozens were arrested across the city during the clashes, which came with the international spotlight shining brightly on France as host of the Euro 2016 football championships.
Police said 29 members of the security forces were among those injured in Paris, while three cars were burned on the city streets.
Prime Minister Manuel Valls blasted the troublemakers, who he said wanted to "kill" the police officers patrolling the demonstration and accused the hardline CGT union, which organised the marches, of an "ambiguous attitude" towards those attacking the authorities.
"I call on the CGT not to organise this type of demonstration in Paris any more," Valls said.
Two further protest days are set for later this month.
Police and organisers gave wildly different figures for the turnout, with unions saying 1.3 million people had turned out across France but police estimating the crowds at 125,000 or more, some 80,000 of them in Paris.