“I am quite clear, not a good result for the Labour Party,” Mr Gilmore said.
“I don’t intend to ignore it, I don’t intend to ignore what we have heard from the people in Meath and we will address it.”
Labour’s defeat was compounded by the success of its senior coalition partner Fine Gael, whose candidate Helen McEntee won the Dáil seat formerly held by her late father Shane McEntee.
Speaking to Kfm, Labour Party Chair Jack Wall says the Programme for Government should be re-examined, but no one has come to him calling for Mr Gilmore's resignation.
"Certainly, I haven't had a rush of people of people coming in to tell me that Eamon Gilmore should go, or that he should consider his leadership, and that's an honest reply."
Mr Gilmore refused to be drawn on whether his leadership of the party would be challenged.
Instead, he said, Labour would tackle its popularity slump the same way it addressed the country’s financial woes.
“The responsibility of leadership is to face down a challenge just as two years ago we faced into the challenge of dealing with the economic financial crisis the country was faced with,” Mr Gilmore said.
“We are going to face the electoral difficulties that the Labour Party now faces.”
The party was eliminated from the by-election running in the second count yesterday, after Mr Holmes secured only 1,245 votes.
He was left trailing behind Ms McEntee and her closest rival, Fianna Fáil’s Thomas Byrne, who won around 39% and 34% of the votes respectively.
Labour also lost out to Sinn Féin candidate Darren O’Rourke and Ben Gilroy of Direct Democracy Ireland.
Speaking at the count in Ashbourne, Communications Minister Pat Rabbitte said the party might have to coax Fine Gael into renegotiations of the Programme for Government.
Recent opinion polls have revealed a gradual waning in Labour’s popularity.
The party has been accused of breaking its pre-election promises and abandoning its left-wing policies as part of the Government’s much-criticised austerity measures.
These have included the incoming property tax and cuts to child benefit and education.
Mr Gilmore gave his first public reaction to the disastrous by-election as he attended the Unitarian Church in Dublin to mark Good Friday.
The Tánaiste opened the church’s annual service to commemorate the more than 3,500 people who died as a result of the conflict in the North.
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