Cologne police responded with tear gas and water cannons after right-wing groups threw beer bottles, firecrackers and stones at officers in riot gear. Several officers and a freelance journalist were injured, police said, though the extent of their injuries were not known.
Fifteen people were arrested but that number may grow as videos are viewed, police said.
Many of the protesters were angry at police response to the attacks as well as the influx of migrants and refugees into Europe.
"Where were you on New Year's Eve?" one protester yelled at police "Why didn't you protect those women?"
Thirty-one people, most of them North African or Middle Eastern countries, have been charged in the attacks. Of those, 18 have been identified as asylum-seekers.
Cologne police said they received a total of 379 complaints about New Year's Eve in Cologne, with about 40% being investigated as sex crimes. Other European cities reported similar rashes of sex crime reports.
German Chancellor Angela Merkel has condemned the attacks in Germany as "disgusting, criminal acts" deserving of a decisive response. But she did not back down on her commitment to welcome refugees who obey German laws and pledge to integrate into German society.
In Cologne, where many of the attacks took place, a police spokesman confirmed that Chief Wolfgang Albers was fired Friday. Albers' dismissal came amid criticism of his department's handling of the violence.
One victim of the Cologne violence told CNN there were too few police on the streets to prevent attacks.
"We ran to the police. But we saw the police were so understaffed," the victim said. "They couldn't take care of us and we as women suffered the price."
Spiegel Online reported that groups of men prevented officers from reaching those crying out for help.
"The events of New Year's Eve like a spotlight once more highlighted the challenge we face in a new aspect that we had not really looked at before," Merkel said Saturday in Mainz, at a meeting of her conservative CDU Party.
Merkel garnered international acclaim for her decision to welcome many of the hundreds of thousands of Middle Easterners who have made the journey to Europe this year in search of safety, food, work and a better life.
But Merkel stressed Saturday that Germany's welcome was not unconditional.
"We, of course, expect from the refugees who come to us, those seeking protection, that they have the will to integrate, that they strive for integration," she said.
Her response showed a determination to avoid repeating what are broadly viewed within Germany as the mistakes officials made with regard to Turkish "guest workers" after World War II. Because it was assumed that arrangement was temporary, virtually no attempt was made to learn (or teach) the language, customs and mores of the new home country.
The mistakes of that era echo through Germany to this day. Generations later, some families of Turkish origin living in Germany do not speak German, and forests of satellite dishes are all aligned to receive Turkish TV broadcasts. Many young third- or fourth-generation Turks lack a sense of belonging, a sense of being German, even though their grandparents were born in the country.
Cologne mayor criticized for advice to avoid men
And Merkel continued to say the newcomers were welcome.
Tough prosecution of wrongdoing, she said, "is in the interest of citizens of Germany, but it is just as much in the interest of the large majority of refugees who are here with us and, therefore, it is absolutely right."
Those who break German laws may forfeit their right to residency and to asylum, she said.
Germany is going through experiencing recriminations over who should have prevented the attacks and whether someone was asleep at the wheel.
Interior Minister Thomas de Maiziere has criticized the response of Cologne police, and German Justice Minister Heiko Maas was among many who expressed disapproval of Cologne Mayor Henriette Reker for advising women to keep "more than an arm's length" away from unknown men.
Reker later said the comments had been taken out of context.
Cologne police spokeswoman Christoph Gilles told reporters Friday that 170 criminal complaints had been filed related to the apparently coordinated attacks, "at least 120 of which have a sexual angle."
An 80-person investigative team is looking at 250 videos with about 350 hours of footage, Gilles said.
The 31 suspects charged include nine Algerian nationals, eight people from Morocco, five from Iran and four from Syria, German interior ministry spokesman Tobias Plate said. Two are German citizens, while one each comes from Iraq, Serbia and the United States.
Other German cities experienced similar attacks the same night, including the northern city of Hamburg, where more than 50 incidents were reported.
Attacks reported in Sweden, Switzerland, Austria and Finland
Similar incidents occurred elsewhere in Europe.
Police in Kalmar, Sweden, said Friday they had arrested two men described as refugees who only spoke Arabic or Kurdish. The men, who have since been released on bail, are suspected in two of the 11 sexual harassment complaints received in the town on New Year's Eve, according to police spokesman Johan Bruun.
Police are looking for 10 to 15 additional suspects who allegedly encircled a group of women in Kalmar and groped them, Bruun told CNN.
In all, 16 females between 17 and 21 said they were molested, he said.
In Salzburg, an Austrian city on the German border, two men have been charged with sexually assaulting women on New Year's Eve and New Year's Day, police said in a statement.
A 23-year-old Syrian citizen was arrested in the city's Old Town section on New Year's Eve after sending fireworks in the direction of a 17-year-old student, police said. She was slightly injured in her abdomen. Further investigation revealed he allegedly had sexually abused a 20-year-old from Bayern in the inner city before the fireworks incident, police said.
A 28-year-old Afghan allegedly sexually abused a 58-year-old woman in the early hours of January 1, police said.
Another woman reported she was sexually harassed on New Year's Eve by two men who also stole her phone, police said. A 24-year-old Afghan man was arrested and accused of molesting a 26-year-old January 6 at a downtown Salzburg pub, police said.
In Zurich, Switzerland, six women told authorities they were "robbed from one side, [while] being groped ... on the other side" by groups of men described as having dark skin, according to a Zurich police statement released Friday.
And in Helsinki, Finland, police said they are investigating two possible criminal offenses related to New Year's Eve harassment centered around "a gathering of asylum-seekers."
Both the Zurich and Helsinki allegations became public well after the incidents took place.
German official: Asylum-seekers convicted of attacks can be deported
Critics had chastised authorities and media in Germany for not being upfront earlier about the attacks there -- though that news did get out sooner, spurring anti-immigrant protests and challenges of Chancellor Angela Merkel for her openness to refugees from war-torn nations like Syria and Iraq.
Maas, the German justice minister, said Thursday that anyone convicted of such crimes could be deported
if they were seeking asylum.
"The law allows for people to be deported during asylum proceedings if they're sentenced to a year or more in prison, and that's possible with sexual offenses," Maas said.
Germany's embrace of migrants challenged
The idea that some refugees fleeing poverty and violence would commit crimes in the countries that welcomed them has spurred anger across Europe and calls for action.
Merkel has been praised
in many circles for her not only urging Europe to find a place for such migrants, but backing up her words inside Germany. The nation took in about 800,000 migrants last year, far more than any other European nation.
But critics have questioned the ability of so many migrants, predominantly Muslims, to suddenly adapt to a European way of life, and the apparently coordinated attacks in Cologne and elsewhere in Germany have fueled that sentiment.
Slovakian Prime Minster Robert Fico called for a summit of European Union leaders to discuss the refugee situation, and said "the idea of a milticultural Europe is unrealistic."
"It's a fiction, it's a dream. Simply, it doesn't work," he said.
While officials warned against scapegoating all migrants, Merkel herself has been vocal in slamming the "intolerable" attacks and indicating her government will send a "clear sign" to those who don't respect German law.
She also suggested what happened New Year's Eve may be symptomatic of a broader problem with regards to endangering women.
"I don't think these are single cases," Merkel said Thursday. "People have a right, and we as a state ... have the obligation to give the right answers to this."