Drunk fans damaged buildings and threw bottles at riot police on Wednesday night before hundreds clashed with police at the Spanish Steps on Thursday before the game.
Police fired tear gas and tangled with fans, who left the square's Baroque fountain full of empty beer bottles.
"Rome has been violated today," local government councillor Paolo Masini said in a statement, describing tourists holed up in hotels and nearby windows shuttered. "This is pure and simple hooliganism."
Italian police said they arrested six Dutch nationals on Wednesday night and confiscated truncheons and broken bottles. Judicial sources said a further 17 people were arrested on Thursday, and magistrates will deal with the four not yet charged next week.
Livio Ricciardelli, a member of the city government, said Feyenoord should make a public apology and pay for the damage caused to the city by its fans, adding that repairs would be costly.
The Dutch embassy in Rome said in a statement that there was "no place for violence in football" and that it would cooperate with Italian authorities to punish the perpetrators.
Italian Prime Minister Matteo Renzi said during a television interview that he appreciated the embassy's message about the events, and would take the issue to European football authorities, calling it "an insult to civilised society."
"Tomorrow we will check how it worked, if we made any mistakes, on the understanding that these hooligans are evidently responsible," Renzi said in response to a question about the incidents on state news channel RAI2.
Most of the charges against fans are for resisting arrest, judicial sources said, adding that the first eight charged were given six months' jail time, or they can pay a 45,000-euro (33,175 pounds) fine.
Rome's public works department said staff had been sent to empty and clean the Barcaccia fountain, a boat carved out of travertine stone that nestles in an oval pool at the foot of the Spanish Steps.
Technicians will now assess what damage has been done to the recently restored fountain, which was designed by Italian artists Pietro and Gian Lorenzo Bernini.
Around 6,500 Feyenoord supporters were expected to attend the last 32, first leg match at the Stadio Olimpico.
(Additional reporting by Jacopo Lo Monaco; editing by Justin Palmer and Matthew Lewis)
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