Arlene Foster has called on Michelle O'Neill to show "contrition" as pressure mounts on the Deputy First Minister ahead of a party leaders' meeting at Stormont on Friday.
The five other main political parties in Northern Ireland are all calling on Ms O'Neill to step aside over her actions at Bobby Storey's funeral.
But Mrs Foster told the Belfast Telegraph that, despite growing public anger over the coronavirus guideline breaches, she would not be collapsing the Executive if Ms O'Neill refuses to resign.
The Ulster Unionists have said that an apology isn't enough and that Ms O'Neill has damaged public confidence in Stormont so much during a major health crisis that she has to go.
The First Minister was speaking as details emerged of a stormy 90-minute Executive meeting on Thursday during which ministers from every other party rounded on Ms O'Neill.
"It was relentless," said a Stormont source.
"Michelle was hammered from all sides over her unacceptable behaviour at the funeral. She continued to insist that she had done nothing wrong."
A meeting of the party leaders' forum - the second since devolution was restored six months ago - was called by Mrs Foster to address the crisis.
It will be held at 10am at Stormont. The SDLP, Alliance, Ulster Unionists and the Green Party are also calling on Ms O'Neill to stand aside. People Before Profit is the only party not doing so.
Sinn Fein president, Mary Lou McDonald, said no-one should be "punished" for attending the funeral of a friend.
But under Stormont guidelines, friends of a deceased person should attend the funeral only if none of the bereaved family members will be present. Mr Storey's family did attend Tuesday's funeral.
Mrs Foster said: "Michelle needs to seriously reflect on the hurt, anger and upset that has been caused by telling people to abide by rules which she has not done herself.
"It is very damaging to the public health messaging of the Executive. It causes a major credibility problem."
The joint Executive press conference hosted by the First Minister and Deputy First Minister was cancelled on Thursday.
Mrs Foster said: "I can't currently stand beside Michelle and give out health advice.
"It can't be business as usual. But the DUP won't bring down the Executive because that would be only punishing the people of Northern Ireland for the bad behaviour of Sinn Fein.
"As a party, we will not be walking away from government during a public health crisis."
However, Mrs Foster said that she was not prepared to appear on a joint platform with Ms O'Neill until the row over her attendance at the funeral was resolved.
Speaking in Dublin, Mary Lou McDonald said that she wouldn't be calling for her colleague Ms O'Neill to step aside as the Executive had a heavy workload to do due to the Covid-19 pandemic.
She said: "That work is under way but it is not finished yet, so, no, I don't think anybody should be punished or asked to step aside or step down because they attended the funeral of a friend.
"In relation to Bobby Storey's funeral, I was very honoured to be asked by his family to attend the mass.
"I was one of a very small number of people to be there and to do a reading at that mass. There was a limited cortege and people lining the streets.
"But this is the problem, when somebody is a public figure or where somebody dies in very tragic circumstances - people will come out to pay their respects."
Ms McDonald claimed that the funeral organisers "did their best to maintain social distancing".
The DUP has asked the PSNI to investigate the Storey funeral breaches.
The party's South Antrim MLA and vice-chair of the Health Committee, Pam Cameron, also called on all the Sinn Fein MLAs who attended the event to self isolate to ensure that they are not spreading Covid 19.
Ms O'Neill has defended her actions, insisting she acted within Covid-19 rules.
She has differentiated between the management of the funeral cortege and service and scenes along the route when thousands gathered to watch the procession pass. The Deputy First Minister has insisted that was outside of her control.
But Ulster Unionist leader Steve Aiken rejected her arguments. "There has to be a mea culpa from Michelle O'Neill. She has to say what she did was wrong and she has to step aside," he said.
"That is the only way to restore faith in the Northern Ireland Executive at a time when potentially thousands of people's lives are at risk from coronavirus.
"In any normal political society, Michelle O'Neill would be gone by now.
"This is not an orange or green issue. The integrity and authority of the Executive is at stake."