Eight people have been arrested following the Old Firm clash on Saturday.
Reaction To Scotland's Top Cop Quitting
Sir Stephen House's decision to step down early will not solve the "deep-rooted" problems within Police Scotland, politicians have warned.
The chief constable's announcement was described as an opportunity for a "fresh start" for the force after a series of high-profile controversies.
Opposition parties blamed the SNP government, accusing them of pushing through cuts and failing to ensure proper oversight.
Scottish Liberal Democrat leader Willie Rennie called for a wider inquiry into Police Scotland's operation.
He said: "Decapitating the chief constable won't solve the deep-rooted problems in Police Scotland. A new chief carrying on as if nothing is wrong will cement the problems rather than solve them.
"This isn't about the job of one man at the top but recovering the fortunes of the wider police force, which is in the doldrums."
Scottish Conservative leader Ruth Davidson said: "His leadership has been a source of controversy from the outset and Police Scotland needs a fresh start.
"It's essential that his replacement is up to the task of tackling the problems that have afflicted the single force since its inception.
"But let us not forget that the buck stops with the Scottish Government and the Scottish Police Authority on Police Scotland's failings."
Graeme Pearson, Scottish Labour's justice spokesman and a former deputy chief constable of Strathclyde Police, said: "We know the problems in Police Scotland extend far beyond the chief constable.
"Police Scotland has been an organisation without proper oversight for too long and has had to endure cuts from the SNP government which have left the force under-resourced and over-worked.
"Officers and staff work round the clock to keep people safe. The controversy of this summer has dragged their reputation unfairly through the mud."
George McIrvine, branch secretary of Unison police staff Scotland, said: "Sir Stephen House was given the impossible task of merging Scottish policing into one police force while having to meet political targets of 1,000 extra police officers, and at the same time introducing #1.1 billion efficiency savings target. These targets had little to do with good policing.
"Over 1,700 police staff have left their jobs while Police Scotland has been beset with problems.
"Sir Stephen House should have publicly called on the Scottish Government to scrap all political targets and called for proper funding of our police service."
His warning against political interference in policing was echoed by Brian Docherty, chairman of the Scottish Police Federation, which represents rank-and-file officers.
He said: ''Sir Stephen has made a monumental contribution to policing. He has delivered the most significant public-sector restructuring in a generation against a background of a brutal austerity agenda.
''I have little doubt that history will prove to be kinder to Sir Stephen than the current commentary, which at times has been vindictive and deeply personal.
''Many people feared that a single police service could be susceptible to political interference and those who have called for the head of the chief constable as some form of trophy need to consider that.''
Scottish Police Authority chair Vic Emery paid tribute to Sir Stephen as "the right individual at the right time'' to lead Police Scotland.
He said: "Steve has always been a constable first, and a chief officer second. He has continued to see policing as a lifelong personal vocation, not simply a career ladder to climb.
"I believe that will be a view of him shared by many thousands of officers and staff across Police Scotland.
"As Chief Constable of both Strathclyde Police and Police Scotland he has served the people of Scotland with distinction for over a decade."
Justice Secretary Michael Matheson said: "I would also like to thank the Chief Constable for his service and contribution to policing in Scotland.
"There have been challenges but we are learning the lessons and addressing the issues. We continue to make good progress on the reform journey to deliver a truly integrated single service and all the benefits that brings and I will expand on this in a statement on policing to parliament next Thursday.
"The people of Scotland deserve exemplary policing delivered by an exemplary police service and we will continue to work with Police Scotland, the Chief Constable and his successor, and the new chair of the Scottish Police Authority to ensure this is the case.''
A taxi driver has been charged under the road traffic act after a crash in which his three passengers were injured.
A majority of voters north of the border believe the SNP would have the right to hold a second independence referendum if the party wins more than half of the Scottish seats in the General Election, a poll has found.
A bakery has closed temporarily following an outbreak of Hepatitis A in North Lanarkshire.
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