The mother of IRA murder victim Paul Quinn has said that Sinn Fein's Conor Murphy is "unfit for office" and should resign.
Breege Quinn welcomed the Finance Minister's apology for branding her son a smuggler and a criminal after his brutal killing in 2007.
But she said it was unacceptable that he had "maintained an untruth for 13 years".
Sinn Fein president Mary Lou McDonald said there was "no question" of Mr Murphy stepping down.
The Ulster Unionists and the TUV have called on him to resign, but the DUP has declined to do so.
On Wednesday night Mrs Quinn told the Belfast Telegraph that Mr Murphy's words blackening her son's name had torn her family apart.
"We are glad that Paul's name has finally been cleared by Sinn Fein," she added.
"We knew that he wasn't a criminal. Conor Murphy knew that too. He knew in his heart and soul that what he said wasn't true, but he never retracted it.
"We've asked him to do that since 2007, and Conor and Sinn Fein ignored us. The party changed its tune because it was caught in a corner with the Dail election. I welcome Conor's apology but I think that he must stand down.
"If he branded Paul a criminal in the spur of the moment, he should have retracted in the days after. He chose not to. It took an election 13 years later for him to withdraw his words blackening my son. That makes him unfit for office."
I very much regret comments I made in the aftermath of Paul's murder which have added to the grief felt by the Quinn familyConor Murphy
Paul Quinn was beaten to death by an IRA gang in a barn in Oram, Co Monaghan.
Mr Murphy previously denied calling the 21-year-old victim a criminal.
But he apologised yesterday after footage was uncovered of him making the slur in a BBC NI Spotlight interview.
Speaking to RTE, Mr Murphy said he had "consistently and unreservedly condemned the murder of Paul Quinn".
He said those responsible were criminals and should be brought to justice.
He also appealed to anyone with information on the murder to go to the gardai or PSNI.
"I very much regret comments I made in the aftermath of Paul's murder which have added to the grief felt by the Quinn family," Mr Murphy said.
"I apologise for those remarks and I unreservedly withdraw them.
"Once again, I offer to meet the Quinn family at a time and place of their convenience."
The Quinns said they would consider Mr Murphy's offer to meet them over the next few days.
Ms McDonald said she would not be asking him to resign.
"Absolutely not. We just got power-sharing up and running and I believe that everyone concerned is determined to make a success of it," she added.
"Conor has clarified the fact that he is withdrawing his statement. (These were) comments he made over a decade ago and he has apologised for them. That's the right thing to do. It's the decent thing to do.
"I know that he will be hoping to meet with the Quinn family to talk to them.
"That family have been through a horrible ordeal.
"I mean, I can't even imagine... to lose their son in such a brutal way."
Asked if she was happy to continue working with Mr Murphy, First Minister Arlene Foster said Executive appointments were made by individual parties, so his future as Finance Minister was "a matter for Sinn Fein".
She added: "Wherever possible, if mistakes are made we should reflect.
"I haven't heard Conor's apology, but I think it is right that should happen.
"I'm sure Mr and Mrs Quinn would want above all to have justice for their son."
However, Ulster Unionist leader Steve Aiken called for the Finance Minister's resignation.
"In any normal society, a senior minister who disparaged the victim of a callous murder, (that) would result in their resignation or sacking," he said.
Ulster Unionist MLA Doug Beattie added: "Every day he remains in office is an insult to the Quinn family and anyone who respects the rule of law and democratic politics.
"Sinn Fein appointed Conor Murphy as a minister and they can remove him."
He should resign. In any system other than Stormont he would be goneTUV leader Jim Allister
SDLP leader Colum Eastwood raised the matter with Prime Minister Boris Johnson in the House of Commons.
He said Mrs Quinn's "fearless pursuit of those responsible for the horrific murder of her son stands in stark contrast to the cowardice of those involved and those who have sought to damage his character and blacken his name".
He added: "They broke every bone in Paul's body.
"And for 13 years they have tried to break his family's resolve."
TUV leader Jim Allister called on Mr Murphy to resign.
He said the Executive could not ignore the magnitude of what had happened.
"The issue is not something to be swept under the carpet. The Finance Minister has been irredeemably exposed as unfit for office," Mr Allister said.
"He should resign. In any system other than Stormont he would be gone."
Alliance deputy leader Stephen Farry said: "While a public apology is the correct thing to do, it is disappointing it did not happen at any point since Conor Murphy's comments but rather was left until the wake of a major election in which Sinn Fein are coming under pressure.
"Anyone with information on those who carried out the brutal killing of Paul Quinn need to cooperate fully with the police investigation."
The courageous stand of Breege Quinn, not just in the last few days but over the past 13 years, as anyone who has followed her activity on social media will know, is an indictment of a political class and a compliant media who have invested so much in a process which requires victims be ignored in order for it to survive.