First Minister Arlene Foster has led calls for an apology from Michelle O'Neill, who was among senior Sinn Fein members attending the funeral of leading republican Bobby Storey on Tuesday.
Mrs Foster said the Deputy First Minister had "undermined" the Executive.
"She needs to apologise, she needs to recognise the wrong that has been done and she absolutely needs to make amends for what has happened," she said.
Alliance leader Naomi Long, UUP MLA Doug Beattie and TUV leader Jim Allister all said Mrs O'Neill should consider her position.
And there have now been calls for Sinn Fein politicians in the Republic who attended the funeral with thousands of mourners - including new leader of the Opposition Mary Lou McDonald and TD Pearse Doherty - to stay away from the Dail for two weeks in quarantine.
However, Mr Doherty said he would not advise anyone attending the funeral to quarantine.
"Neither would I encourage people who went to Tesco yesterday to quarantine either," he added.
Fine Gael Senator Garret Ahearn has written to the Dail chairman saying such a request is "in the interest of everybody's health and safety".
"It would be extremely regrettable if a cluster of Covid-19 cases was created in Leinster House because some members travelled to Belfast," he said.
Meanwhile, a former Sinn Fein political adviser said watching the scenes from the funeral had left her "absolutely fuming".
Naomi Bailie had served as an adviser to Chris Hazzard MP.
"What happened there was totally indefensible and to hell with any of them who try to spin it," she said in a social media post. "If there's a straw to break the camel's back in my case, this is my back broke tonight."
Justice Minister Mrs Long said when the rule makers break the rules it is more hurtful "for all who made huge sacrifices to obey the regulations".
To be frank, it broke our hearts.Helena Dallat-O'Driscoll
While the UUP's Doug Beattie said Mrs O'Neill had "absolutely damaged the office of the Executive by her actions".
SDLP leader Colum Eastwood said the display "showed a blatant disregard for the sacrifices made by so many".
One of the families to have made that sacrifice was that of his party colleague John Dallat MLA, who died on May 5.
"It has been very, very painful and this has compounded that pain," his daughter Helena Dallat-O'Driscoll said.
"When we were organising dad's funeral we did what we thought we needed to do. We had no wake. We didn't get to bring his remains home. We had a very small, private funeral. We didn't carry his coffin. None of his MLA colleagues were able to travel to it.
"To be frank, it broke our hearts. We felt we had to make those sacrifices in the greater interest of safety and society. As a family we didn't think we should be treated any differently."
She added: "I think it's obvious Michelle O'Neill needs to reflect on how that funeral appeared to be conducted. She needs to reflect on her authority to give advice to the rest of us on Covid restrictions."
In a statement last night Mrs O'Neill defended her attendance.
"The organisers of Bobby Storey's funeral put in place measures to ensure compliance with the public health guidelines and regulations around funerals," she said.
"This proactive approach by the organisers included a cortege limited to 30 people, a socially distanced Mass, and stewarding along the route to prevent crowds joining the cortege.
"The organisers also live-streamed the funeral to encourage people to stay at home and pay their respects.
"In advance of the funeral, the organisers were in contact with the PSNI and made them aware of the arrangements. I am confident that the organisers ensured the funeral Mass, cortege and ceremony was within public health regulations and guidance."
The Diocese of Down and Connor said its churches had been advised on June 4 that numbers attending funeral Masses should be restricted to 10 mourners.
The diocese said it received updated guidance from The Executive Office on Tuesday evening - after the funeral had taken place.
"This guidance was communicated to priests, funeral directors and all involved in the organisation of funerals," it said in a statement. "It is for the statutory authorities to investigate suspected breaches of the Health Protection (Coronavirus Restrictions) Regulation NI 2020. Since a PSNI inquiry is under way, it would be inappropriate for the diocese to make any further comment."
Earlier, Secretary of State Brandon Lewis questioned the attendance of Sinn Fein Executive ministers. Mr Lewis said he understood that some people were "frustrated and angry".
"People have given so many sacrifices over the last couple of months," he said. "I can understand people's frustrations. It's not something I would have done."