First Minister Arlene Foster has said she cannot currently “stand beside” Michelle O’Neill and “give out public health advice” after the deputy first minister’s attendance at a funeral.
Mrs Foster said the executive’s credibility had been damaged and she should stand down while investigations take place.
Ms O’Neill has defended attending the funeral of Bobby Storey.
Sinn Féin has said she will not stand down “under any circumstances”.
The party said the funeral was organised in accordance with the “grieving family’s wishes”.
Ms O’Neill said her actions were in line with social distancing and accused her critics of political point-scoring.
“The organisers did everything they could to ensure regulations were met and if it was not for the plan that organisers put in place with the PSNI to limit numbers then there would have been even bigger numbers in attendance,” a Sinn Féin spokesperson added.
However, on Thursday evening, Mrs Foster, the DUP leader, said Ms O’Neill “stood with me on Monday and made pronouncements about public health advice and on Tuesday she went ahead and broke that health advice”.
“So I think there is a credibility issue here and I said yesterday that she should reflect on that, she should apologise.
“She chose not to, instead doubling down on the fact that she had done nothing wrong and I think the wider community in Northern Ireland want to see action in relation to this.”
Mrs Foster added: “That is why I have now said she should consider her position and she should think about stepping aside whilst all of the investigations take place into what has become a real credibility issue for the Stormont executive.
“I cannot stand beside Michelle currently and give out public health advice, because people will just say there’s no credibility to that given what has happened this week.
“It cannot be business as usual when people are saying to me there is no credibility to your message.”
Speaking in Dublin, Sinn Féin president Mary Lou McDonald said Ms O’Neill should not stand down.
“I don’t think anybody should be punished or asked to step aside or step down because they attended the funeral of their friend,” she said.
New guidelines issued
Current coronavirus regulations state a maximum of 30 people are allowed to gather together outdoors.
About 120 mourners were inside St Agnes’s Church in Belfast for Mr Storey’s funeral, BBC News NI understands.
The deputy first minister told a Stormont committee on Wednesday that Requiem Masses were now possible due to this week’s reopening of places of worship.
However, guidance on religious services issued to faith leaders last week stipulated that it did not apply to weddings, baptisms or funerals.
Updated guidance, seen by BBC News NI on Wednesday and published on the Department of Health website on Thursday, said funeral services could now be conducted in a place of worship.
“The size and circumstances of the venue will determine the maximum number that can attend the service safely whilst observing social distancing of at least 2 metres, wherever possible,” it continued.
“It is recommended that face coverings are used for indoor services.”
The guidance for those gathering outdoors remains the same with the guidance, issued for funeral directors, stating “a maximum of only 30 are permitted to gather for the committal at the graveside or at the front of the City of Belfast Crematorium”.
It is a crisis, but we are not yet talking about Stormont falling - the DUP has said that it will not be bringing it down.
They have demanded an apology, and the temporary stepping aside of Michelle O’Neill for attending Bobby Storey’s funeral and potentially breaching guidelines.
The Sinn Féin statement made it clear that she is not going to step aside under any circumstances.
There will be a meeting of the party leaders’ forum, in which the other four party leaders may tell Michelle O’Neill that she has broken these regulations, and call for her to step aside.
In terms of anything further, I would be surprised if the PSNI would get into the heart of such politically difficult terrain by bringing charges against the likes of Michelle O’Neill and any others in the Sinn Féin leadership.
In terms of the assembly process, we don’t even have a standards’ commissioner at the moment to examine whether she broke the assembly’s standards - because that was one of the positions allowed to go vacant during the three-year Stormont stand-off.
SDLP Leader Colum Eastwood told The Nolan Show that Ms O’Neill should “absolutely step aside” following the funeral.
Mr Eastwood said Sinn Féin was “putting themselves ahead of the people”.
“People buried children during this (lockdown) and weren’t able to have a proper send-off,” he said.
“I think it is reprehensible,” he added.
All of the other parties which make up Northern Ireland’s devolved government along with Sinn Féin - the DUP, UUP, SDLP and Alliance - have now called on the deputy first minister to step aside or to resign.
Speaking on BBC NI’s Good Morning Ulster programme, UUP MLA Doug Beattie said the deputy first minster “is in absolute denial if she thinks she adhered to the advice she has given”.
“The images show the absolute opposite, she did not adhere to the advice she told the rest of the public to adhere to and many people did adhere to,” he said.
The leaders of the five executive parties are to meet on Friday.
Criticism of the crowd
Mr Storey was considered the head of intelligence of the IRA for a period from the mid-1990s.
Tuesday’s funeral was the latest in a number of events that have been criticised for attracting crowds during the Covid-19 pandemic.
In May, a senior police officer said there had been social-distancing breaches at funerals in both the unionist and nationalist communities.
Earlier in June, protestors at Black Lives Matter rallies in Belfast and Londonderry were fined, while several hundred people took part in a “save our statues” rally at Belfast City Hall.