The Irish Road Victims Association (IRVA) has called for the Agriculture Minister to step down from his role.
The newly appointed Agriculture Minister acknowledged it was a serious lapse of judgment.
Mr Cowen said he was embarrassed after it emerged last week that he was barred from driving for three months in 2016 after he was found to be drink-driving.
He was stopped at a checkpoint and breathalysed on his way home to Co Offaly after an All-Ireland football final in Dublin.
"I think his position is untenable, as much as I don't like to kick someone when they are down and when they have made an apology, but it really goes further than that," said Donna Price, founder of the IRVA.
"It shows a complete disregard for other road users.
"I'm sure that the fact that he was caught on this occasion that it wasn't his first occasion of drink-driving so I doubt very much that he was just unlucky on this one occasion."
The Taoiseach believes Minister Cowen should not be punished for something that happened years ago.
Speaking to Red FM, Micheál Martin said: "Are you going to condemn someone for the rest of their life for something that might have happened four or five years ago?
"I think there is an issue of proportionality in that regard."
Justice Minister Helen McEntee has said that Mr Cowen was "wrong" but added she has accepted his apology.
Ms McEntee, who was visiting Slane Garda station in Co Meath where she held a meeting with Garda Commissioner Drew Harris, said the incident "shouldn't have happened".
She added that the focus of the new Government should be on road victims and their families.
"What happened was wrong. It shouldn't have happened and I think the minister has acknowledged what he did was wrong and I think what's important at the time is that the law was applied," Ms McEntee added.
"Obviously what was given out to the minister, he accepted it and took that as he should have.
"I think we need to focus on victims, those who have lost family members due to drink-driving, making sure that we have measures in place that protect people, that make our roads as safe as possible.
"Obviously there was a huge amount of changes implemented in the last government and we want to continue to implement them.
I don't agree with what happened, he has apologised, I accept that and I think it's important that the law was applied.
Ms McEntee said Mr Cowen is to make a statement to the Dáil and answer questions on his driving ban.
Mr Cowen, who was appointed minister as part of the coalition deal to form the new government, said over the weekend: "Maybe I should have stood up in the Dáil, and led from the front and acknowledged my stupid mistake.
"I was not trying to keep it secret, I was trying to get on with my job."
Additional reporting by Digital Desk