Fianna Fáil minister Stephen Donnelly says he has full confidence in Micheál Martin as party leader, and believes "internal bickering" is bad for the party .
His comments follow a call from Barry Cowan for a special meeting to analyse the party's poor performance in the Dublin Bay South by-election.
The party's candidate Deirdre Conroy finished with less than 5% of first preference votes - putting her in fifth place.
Sligo-Leitrim TD Marc MacSharry has called on Mr Martin to resign as party leader during the upcoming Dáil summer recess.
— Marc MacSharryTD (@MarcMacsharryTD) July 11, 2021
However, Minister Donnelly says he "absolutely does not" agree the Taoiseach should step down.
He said: "I have full confidence in Micheál Martin - I think he's doing a really good job in what is an incredibly difficult period for the country.
"I think the sort of internal bickering is bad for the party - I think the party has an incredibly important job right now in Government.
"We have to deal with COVID, and make sure the vaccination programme runs through. We have a very ambitious programme in place for reform in healthcare, and really serious targets on housing. That needs to be the focus right now."
The Health Minister congratulated Labour's Ivana Bacik on her by-election win.
He said Fianna Fáil's result was disappointing, but suggested the record shows Government parties don't win by-elections.
He said the party will now "reflect" on the result, but added his own focus is currently on COVID.
Minister Donnelly's Cabinet colleague Michael McGrath has also voiced support for Mr Martin today.
He said the by-election result was disappointing for Fianna Fáil, but added that "this is not a time for us to be talking about ourselves".
6/6 For Fianna Fáil, all of this is an outstanding opportunity and a monumental challenge. To succeed, we need to support our Taoiseach, our team, work together, have unity of purpose and show the public that their concerns are our priorities at all times.
— Michael McGrath (@mmcgrathtd) July 11, 2021