Micheal Martin said he is “heartened” by the support he has received in recent days despite some members of his own party raising doubts about his leadership.
The Fianna Fail leader said his focus as Taoiseach is on the big issues facing Irish people.
The party faced a disastrous outcome in the Dublin Bay South by-election after their candidate Deirdre Conroy polled under 5% of the votes.
Mr Martin said the outcome of the by-election was a reflection of the support for Ivana Bacik.
The Labour party candidate was elected on the ninth count at the by-election on Friday, while Fine Gael received 26% of first preference votes, and Sinn Fein 15%.
“I have been very heartened by the feedback I have received from the membership of the party who voted overwhelmingly for Fianna Fail to go into Government with a clear purpose to focus on the needs and concerns of the Irish people,” Mr Martin added.
“We are coming through a phase of an unprecedented pandemic.”
He added: “In terms of the Dublin Bay South election, I have made it very clear my views.
“In terms of in the last 30 by-elections, 27 were won by the opposition.
“I think (the by-election) was a personal victory for her (Bacik), given her commitment throughout the years as a parliamentarian and activist.
“That was very clear to me and that’s was the fundamental issue at play.”
The result in the by-election has prompted speculation over Mr Martin’s future, with TD Jim O’Callaghan casting doubt on his leadership and former agriculture minister Barry Cowen requesting a special meeting of the parliamentary party.
Mr Martin said he is “very open” to a meeting of the parliamentary party.
My focus as Taoiseach has to be on the big issues facing the Irish peopleMicheal Martin
“It won’t be this week for obvious reasons and I think we have to take stock of that and the party will discuss that,” he added.
“My focus as Taoiseach has to be on the big issues facing the Irish people.
“Irish people respect that from those in Government.
“I always work with my colleagues, I always engage with my colleagues, even with those who obviously have different perspectives to me and do criticise me.
“I will engage with all colleagues about issues.”