Police Scotland said Rangers supporters had "damaged the image and reputation of Glasgow" as five officers were injured in "ugly" fan celebrations of their team's first premier league title in a decade.
They warned that "many more arrests will follow" after 28 were apprehended as Rangers fans attacked each other and threw missiles at officers - with one suffering a serious face wound.
Police described "disgraceful" scenes as supporters converged on Glasgow's George Square despite being warned not to.
As Glasgow was dealing with a surge in Covid infections and a potential extension of lockdown, police said Rangers fans "chose to ignore" warnings against large gatherings and took to the streets.
Thousands had to be dispersed by police over what police described as "rising disorder" during Saturday afternoon, with fans setting off missiles and flares and drinking in the street to commemorate their club's Scottish Premiership title triumph.
The march from Ibrox Stadium to George Square on Saturday took place the day after Nicola Sturgeon announced that Glasgow would remain in level three of Covid controls for another week because of surging infection rates believed to be related to the variant first detected in India, as well as a travel ban on residents leaving the city.
There had been a high, visible police presence throughout the day as huge crowds gathered outside Ibrox before the final game of the season against Aberdeen.
Fans congregated in George Square throughout Saturday afternoon despite warnings about rising case numbers, and footage later showed the area strewn with broken glass and debris. There was a heavy presence of police and emergency services throughout the day.
Police broke up the crowds after about five hours due to "rising disorder", with one officer suffering a serious facial injury.
Police Scotland Deputy Chief Constable Will Kerr said investigations were continuing and further arrests would be made in the coming weeks.
Assistant Chief Constable Gary Ritchie said: “I recognise that Rangers fans would have wanted to celebrate their club’s success on Saturday and it was anticipated that crowds would congregate.
“Police Scotland, our partners and Rangers Football Club had all asked fans not to gather and to take personal responsibility for their actions.
“But 15,000 people chose to ignore that and took the selfish decision to gather at Ibrox and then George Square, and a robust partnership response plan was implemented to manage them."
Police Scotland said it had declined a request to facilitate the march to George Square and urged the crowd to go home.
After supporters celebrated in George Square for about five hours, officers said some fans became unruly.
Property was damaged, several people assaulted and missiles and flares were thrown at officers.
“There is no easy way to stop that number of people who are intent on coming into a city from doing so without causing significant disruption to everyone else," said Det Chief Cons Kerr.
“Our policing approach will always be to manage a crowd in a situation like this, and minimise disruption to the wider public, while keeping everyone safe.
“The gathering was initially peaceful and throughout the day Police Scotland managed the crowd appropriately, tackling anti-social behaviour and encouraging dispersal under Scottish Government coronavirus regulations.
“However, later the crowd in the square became aggressive and violent, fighting with each other, as the effects of alcohol took hold.
“At this point, public order officers moved in to break up the crowd and make them disperse.
"They began to throw missiles at us and five officers were injured, which is unacceptable. No one should expect to be assaulted when they go to their work."
Police formed a line and cleared the group from the area just after 9pm with Police Scotland describing the fans' behaviour as "disgraceful".
“I strongly condemn the behaviour of these supporters who not only placed our officers at risk but damaged the image and reputation of Glasgow, especially during this critical period of the pandemic," added Asst Chief Constable Ritchie.
“I would like to place on record my thanks to the officers of Police Scotland, especially those who put themselves in danger.
"Throughout the day, the police operation was professional and proportionate.
“So far 28 people have been arrested for a variety of offences but we will be continuing our enquiries and my message is clear that if you have been involved in these ugly scenes then you will be identified and arrested."
Scotland's national clinical director, Prof Jason Leitch said Glasgow was in a "fragile" place and it was "disappointing to see people break the rules, in whatever setting and for whatever reason".
He said: "This virus has not gone away - we're not joking, we're not pretending - it's absolutely real.
"And the risk, particularly now in Glasgow, is quite high and Glasgow is fragile.
"You've seen us make an enormously difficult decision this week on our advice that has kept a bit of Glasgow in the same level.
"It feels like a tough day to watch that yesterday if you are a business owner or a family waiting for lunch or dinner."
In Belfast, police warned they will “gather evidence of potential breaches” of Covid rules as crowds gathered on the Shankill to celebrate the title win.
In a brief statement, the Police Service of Northern Ireland said they are “aware of crowds currently gathering in Belfast’s Shankill area and a number of events are planned for this afternoon".
They added: “Police are in the area and we would remind those involved about the importance of adhering to the Health Protection (Coronavirus) Regulations. We will gather evidence of potential breaches and those responsible for them.
“Investigations will follow and where appropriate, enforcement action may be taken in the coming days.”
First Minister Nicola Sturgeon tweeted: "To say I'm utterly disgusted by the Rangers fans who rampaged through the city would be an understatement."
The first minister added: "In normal times, the violence and vandalism and the vile anti-Catholic prejudice that was on display would have been utterly unacceptable. But mid-pandemic, in a city with cases on the rise, it was also selfish beyond belief."
Ms Sturgeon said she hoped Rangers would "reflect on what more must be done to tackle this behaviour by fans, albeit a minority".
She added: "However, ultimate responsibility lies with those who behaved in such a thuggish, sectarian and selfish manner. And that's why we must let the police do their job."
Scottish Police Federation chairman David Hamilton said the people in George Square had shown a "blatant disregard for the safety of the public and police officers".
He said: "To see that number of people blatantly ignoring all the health advice, the government advice, the police advice - even the club advice to disperse - is just absolutely appalling.
"Policing in this country is, of course, done by consent. The whole strategy throughout the coronavirus pandemic has been about making sure people take personal responsibility and they have not done so.
"These people should not have been there in the first place and when they were asked to move, they should have gone."
On Friday night, scores of Rangers fans lit red flares simultaneously along the River Clyde in the city as they marked the end of the season.
In an earlier statement on Saturday, Chief Superintendent Mark Sutherland, the divisional commander for Greater Glasgow, said: "We understand the importance of football in Glasgow, how it connects our communities and, is for many, a culture and a way of life.
"We are, however, still in the midst of a pandemic and under coronavirus restrictions people should not be gathering for any reason."
He said a request for a fan procession to the city centre had been declined, but police later decided to escort large groups of supporters as they made their way to George Square in order to minimise disruption.
Some arrests were made in the afternoon for anti-social behaviour.
But Chief Supt Sutherland said due to the largely peaceful nature of the crowd at that time "it would not be proportionate for our resources to use a high level of force through public order policing to disperse those gathering".
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