A group which represents families of people killed in road incidents has questioned the veracity of the newly-appointed Agriculture Minister Barry Cowen.
Mr Cowen had a three month ban for drink-driving while on a provisional licence in 2016.
He is set to make a statement to the Dáil on Tuesday on the issue.
Mr Cowen is also facing calls to clarify how long he was driving without a full driver’s licence.
Taoiseach Micheál Martin said on Monday: "He has articulated his profound regret and sorrow and shame at what transpired four years ago.
"I became aware of this on Friday - it's deeply, deeply regrettable, it shouldn't have happened".
Donna Price is chairperson of the Irish Road Victims' Association, and lost her own son Darren in a road accident in 2006.
She told The Hard Shoulder: "It's very upsetting for our families, families whose lives have been absolutely upended and destroyed by drink-drivers or drivers who have otherwise broken the road traffic laws.
"We expect more from those in high office, our public representatives, who we elect to represent us and to legislate for us.
"Minister Cowen he knowingly broke the law, he made a conscious decision to drink and drive that day - he knew he didn't have a licence to drive, quite apart from the drink-driving element of it, and in so doing he's putting all of us in danger."
She said the apology from Mr Cowen "doesn't cut it" for families.
"Our families have experienced the worst, our children are dead, because of the actions of drunk-drivers.
"Many people when they're caught are remorseful, they apologise for their mistake - but it's not a mistake, it's a conscious decision to put others at risk and in jeopardy by your actions.
"Because he's in high office now, he's one of the legislators, we expect more.
"How can you trust somebody like that to introduce road traffic legislation or road safety legislation and support our efforts to introduce effective deterrents to prevent people from doing exactly what he did himself?"
"If you break the law, if you've committed a criminal offence you'll lose your job.
"If it's not the case, it should be the case.
"Certainly our TDs, when they're elected to office, they should have to take some sort of an oath to at least obey the laws of the land - in this case, there were several laws broken".
"Micheál Martin, as well, he has a duty, a responsibility to ensure that the ministers that he employs will introduce legislation to keep us all safe - I'm not sure that I could trust this deputy when he has already made that conscious decision to break the law and endanger all of us.
"By the grace of God he didn't kill or injure someone, he could easily have done so".
"He'd no concern or consideration for his constituents - such as Gillian Tracey, who's lost her child and she's been left with life-altering injuries.
"He'd no concern or thought for families such as hers, or the many hundreds who are members of our organisation whose loved ones have been killed in similar circumstances.
"And I'm afraid we simply can't forgive drunk-drivers, or dangerous drivers or those who drive recklessly".