Taoiseach Micheál Martin has described Minister for Agriculture Barry Cowen's statement to the Dáil last night about his ban for drink driving as an "abject apology".
A spokesperson said it was "clear" that Minister Cowen "deeply regrets what happened", and the Taoiseach "accepts that his remorse is genuine".
However, some Opposition politicians argue that the minister still has questions to answer about the incident in 2016.
In his Dáil statement, Mr Cowen accepted he made a "stupid, stupid mistake" - which had never happened before or since - and he was "profoundly sorry".
He said, what he termed his "grave error", should highlight the consequences of drink-driving and the criticism of him was "fully deserved".
Mr Cowen contended that it was "not uncommon" to drive on a learner permit for years, but accepted it was "bad practice" and he should have "regularised" his position sooner.
The Tánaiste has said he has received an assurance from the Taoiseach that there will not be any more revelations about any more offences involving Mr Cowen that are more serious than the ones we know about.
Leo Vardakar said he is satisfied with that assurance.
He said Mr Cowen had made his apology, given his explanations and taken his punishment.
Mr Varadkar said we need to reflect on the fact that Irish roads are very dangerous places and thousands of people have lost their lives on them.
Most families in Ireland have been affected by someone close to them losing their life on or being injured on the roads, he added, and that is why improving road safety has to be a mission of the Government and would be a key focus of Minister for Transport Eamon Ryan in the years ahead.
Green Party deputy leader Catherine Martin said she accepts "given the circumstances" the contrite apology by Mr Cowen and that he has recognised and learned from it.
Speaking on RTÉ's Morning Ireland, Ms Martin, who is also Minister for Media, Tourism, Arts, Culture, Sports and the Gaeltacht, said he has shown remorse after making a "serious mistake".
However, Ms Martin said that the one issue that was not addressed was why Mr Cowen was driving on a provisional licence and she said he may have to reach out to road safety campaigners and explain that himself.
Susan Gray of PARC Road Safety Group called on the Taoiseach to get clarity from Minister Cowen on outstanding questions about whether he broke other road safety laws apart from the drink driving law he admitted to.
Speaking on RTÉ's Sarah McInerney, she said PARC "is totally not happy" with the minister's explanation in the Dáil last night, in particular in relation to his long term driving on a provisional licence and said there were a number of questions left for him to answer.
Meanwhile, a Sinn Féin source has said Mr Cowen had accepted what he did was completely wrong and apologised; and the matter was now a distraction from urgent issues like affordable housing.
However, sources from the Social Democrats, as well as RISE TD Paul Murphy, said Minister Cowen had yet to answer all relevant questions about the controversy.
Additional reporting Will Goodbody