First Minister Nicola Sturgeon says Aberdeen players "blatantly broke the rules" in the lead-up to Saturday's Scottish Premiership game against St Johnstone being postponed.
Two players tested positive for Covid-19 and six others are self-isolating after coming within "close proximity" in an Aberdeen bar on Saturday.
Sturgeon said she was "pretty furious" about the situation.
Aberdeen say they will not forfeit points because of the call-off.
The game has been rearranged for Thursday 20 August, with their home match against Livingston put back 24 hours to the following Sunday, but it is not yet clear if they will face any other sanction under Scottish Professional Football League rules.
They are due to host Hamilton Academical on Wednesday and are then away to Celtic the following Saturday and Scottish football's joint response group insists those are scheduled to go ahead as planned.
Saturday's game at McDiarmid Park was part of the second weekend of Premiership fixtures after last season was halted by the coronavirus pandemic.
Initially, it had appeared that the game in Perth would go ahead despite Aberdeen being forced to cancel training on Thursday and facing the prospect of playing without eight squad members forced to self-isolate.
However, the postponement came at the request of the Scottish government after a meeting between minister for public health and sport Joe Fitzpatrick, the Scottish FA and SPFL.
"I think that is the right decision," Sturgeon said. "We are expecting members of the public to behave in a highly precautionary manner right now and, when a football club ends up with players infected with Covid - and let's remember this is not through bad luck but through clear breaches of the rules - we cannot take even the small risk that they spread the infection to other parts of the country."
Sturgeon said the Scottish government would be contacting all clubs to emphasise the importance of complying with the guidance and hoped "lessons will be learned".
"Football has been given the go-ahead on the strict condition that clubs and players abide by the guidance that has been agreed. If they don't do that, they put at risk the return of the professional game," she said. "To say this incident is deeply regrettable is an understatement.
"I'm pretty furious about this situation, because it shouldn't have happened. If that guidance is not going to be adhered to then all bets are off."
How have Aberdeen responded?
Aberdeen chairman Dave Cormack said he could "fully understand" the Scottish government had asked for the postponement.
"We are very grateful that the decision has been to postpone, rather than forfeit the game," he said.
"I took the opportunity yesterday to apologise to both football and health authorities and with our fellow Premiership clubs today. We are now dealing with this internally with the seriousness it deserves."
St Johnstone expressed their disappointed at the postponement but stressed that "the health and wellbeing of everybody is of paramount importance in this situation".
All eight Aberdeen players were in a bar linked to the city's spike in Covid-19 cases on Saturday following their opening-day defeat by Rangers.
One of the initial cases was confirmed by a follow-up NHS test and the other player is awaiting further results.
Both are asymptomatic but are self-isolating along with six players who were "in close proximity".
'It's a reminder to us all'- what is everyone else saying?
Celtic manager Neil Lennon: "We made them [Celtic players] aware again this morning. Very concerned. Obviously a very unfortunate incident with serious ramifications for the game."
Hibernian head coach Jack Ross: "It's a reminder of us adhering to the guidelines. Without that adherence we run the risk of obviously situations developing as they have done the last couple of days."
St Mirren manager Jim Goodwin: "We've told the lads that they need to make good choices it's as simple as that. They've got to make sure they don't find themselves in a situation where they are sitting in a pub surrounded by complete strangers."
Kilmarnock manager Alex Dyer: "If it happens again next week we're all going to be in trouble. We have to put it down as a mishap from their [Aberdeen's] point of view."