The Health Minister says he has full confidence in Micheal Martin.
Stephen Donnelly believes internal party "bickering" is bad for Fianna Fáil.
His comments follow a call from party TD Barry Cowan for a special meeting to analyse the party's poor performance in the Dublin Bay South by-election.
Fianna Fáil's candidate Deirdre Conroy won less than 5% of first preference votes, compared to over 30% for the winner - Labour's Ivana Bacik.
Sligo-Leitrim TD Marc MacSharry has also called on Micheal Martin to resign as party leader.
— Marc MacSharryTD (@MarcMacsharryTD) July 11, 2021
However, Minister Donnelly doesn't 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."
He acknowledged Fianna Fáil's by-election result was disappointing, but suggested the record shows Government parties don't win by-elections.
He said the party will "reflect" on the result, but added his own focus is currently on dealing with the COVID-19 crisis.
Mr Martin also has the support of Minister Michael McGrath, who today said this is not the time for Fianna Fáil to be talking about themselves.