A number of people were injured during the annual rally organised by far-right and nationalist movements.
The violence started when groups of youths broke off from the main march to attack a squat and set fires to cars.
They also set ablaze a rainbow-coloured arch symbolising tolerance, and tried to attack the Russian embassy.
The protesters - many of them wearing masks - threw firecrackers at the diplomatic mission. Several youths also tried to scale the building's fence but were stopped by the police.
"There is no justification for hooliganism," Polish Foreign Ministry spokesman Marcin Wojciechowski said.
The authorities also said that at least four police officers were among those injured during the violence.
Thousands of people took part in the main march, which began noisily but peacefully. They carried white-and-red Polish flags and chanted: "God, honour, fatherland!"
For the third year in a row violence broke out during the march, the BBC's Adam Easton reports.
The rally commemorates the day Poland regained its independence at the end of World War One in 1918.
Before that, Poland had been carved up between Russia, Prussia and the Austro-Hungarian empire.
Polish police have fired rubber bullets and tear gas to break up violent clashes during an independence day march in the capital Warsaw.