The Irish Road Victims Association has called for the Agriculture Minister to step down from his role.
Barry Cowen was caught drink-driving while using a learner permit between Dublin and Offaly in 2016.
He was banned from the road for three months, while also had to pay a fine.
He has apologised saying he was embarrassed and humiliated, while the Taoiseach Micheal Martin has said he was disappointed only to learn of the incident on Friday.
Founder of the IRVA, Donna Price, says it is unforgivable.
"I think his position is untenable, much as I don't like to kick somebody when they're down and when they have made an apology, it goes further than that, and it shows a complete disregard for other road users.
"I'm sure that the fact that he was caught on this occasion that wasn't his first occasion drink driving, so I doubt that Minister Cowen was just unlucky on this one occasion".
