The Fianna Fáil leader has said that an interview given by Sinn Féin's Conor Murphy to RTÉ News "adds insult to injury" and amounts to what he called a "qualified" apology.
Micheál Martin said Mr Murphy's comments regarding the death of Paul Quinn amounted to a "less than full apology".
He said Mr Murphy, now Minister for Finance in Northern Ireland, had failed to answer a number of questions fully this evening and he said he should go to the Police with all the information he has.
Mr Martin said Conor Murphy needs to explain why he said what he said at the time when he implicated Paul Quinn in criminality and fuel smuggling.
He said Sinn Fein needs to "come clean" and "accept it did wrong to those families". He said a code of omertà had built up around such issues and needed to be addressed.
Earlier, the Fianna Fáil leader said he would urge everyone to listen to the interview with the mother of Paul Quinn on Today with Sean O'Rourke on RTÉ Radio 1 this morning.
He said Breege Quinn had spoken eloquently about how the Quinn family had been let down over the years.
This afternoon, Conor Murphy apologised to the family of Paul Quinn for comments he made about the 21-year-old after he was murdered in October 2007.
He said the "remarks that I made at the time of his killing are a matter of regret" and he said he wants to "withdraw them unreservedly."
Speaking to RTE News at Stormont in Belfast, the Sinn Féin MLA said he was sorry it "added to the grief that the family have felt" and he said he wants to "apologise to them for that."
He said he hopes to be able to meet the Quinn family in the near future to discuss the matter.
The Sinn Féin leader Mary Lou McDonald had earlier indicated that Mr Murphy would issue a statement in which she said he would "retract and apologise for comments made" in relation to Paul Quinn.
Speaking at the Sinn Féin Head Office in Dublin this morning, Ms McDonald said "the Quinn family have been through a terrible trauma and it is fair to say that the only criminals involved in this scenario are the people who took his life so brutally."
She also said she had made an "honest mistake" when she said she didn't believe the MLA Conor Murphy had said the murdered man Paul Quinn was involved in criminality.
Ms McDonald said she only realized her statement was incorrect when she saw the BBC Spotlight programme where the comments were made.
She said she hadn't recalled the remarks were so pointed, that Conor Murphy hadn't misled her about what he had said and that the comments had been made over a decade ago.
She said she had no difficulty in apologising if anything she said, or failed to say, had caused any anguish to the Quinn family.
Speaking in Lucan today, Fine Gael leader Leo Varadkar said the family of Paul Quinn deserved truth and justice and urged anyone with information about his murder to give it to the police.
Asked if he believed Mary Lou McDonald should ask Conor Murphy to resign as Minister in the Northern Ireland Executive, he said he did not want to politicise the issue.
"This is a mother who's still grieving who was brutally killed at the age of twenty one.
"She's looking for answers, looking for truth, looking for justice and I believe she's entitled to that and those who can give her information, truth and justice should do so.
"I don't want to make this part of the election campaign that's happening at the moment but I think on that particular question of an apology, of information, of truth, of justice, that's what she should get."
The Labour leader said Sinn Féin had to address the Paul Quinn case in an open and transparent way.
Speaking at a campaign event, Brendan Howlin said unfortunately in the past, SF had demonised their victims. He said it was happening with Paul Quinn and had happened also with Maria Cahill.
Referring to last night's debate where Mary Lou McDonald was asked about Conor Murphy's remarks, he said there couldn't be a situation whereby a different answer was given depending on the day the question was being asked.
Additional reporting by Samantha Libreri, Vincent Kearney and Conor Hunt