Taoiseach Enda Kenny has told the Dáil that "there are great sensitivities that need to be carefully considered and there will not be a repeal of the Eighth Amendment in the lifetime of this Government".
Socialist Party TD Paul Murphy said that last Friday's vote in the same-sex marriage referendum marks a move towards a modern, progressive and more secular society.
He said the choice for political parties is how they react to the change that is happening.
Mr Murphy asked for a referendum on the repeal of the Eighth Amendment to end the "barbaric" situation where a woman who terminates a pregnancy even in the case of fatal foetal abnormality faces the threat of 14 years in prison.
The Taoiseach said that Friday's vote demonstrates the importance of politics and he said that a big message from a small country was sent out.
He said that he does not believe the Dáil should be "rushed into a situation" where it deals with something so sensitive as fatal-foetal abnormalities.
He said that in same-sex marriage people could point out exactly what the consequences of such a move would be but that is not possible in terms of repeal of the Eighth Amendment.
Mr Kenny said that referendum on marriage equality was about love, commitment and dedication. He said that a repeal of the Eighth Amendment deals with other elements that are not clear at all.
Mr Murphy said that last Friday marked a cry for a different type of society and the amendment is central to that.
He said that 20,000 women have travelled for abortions in the life of this country.
He recalled Savita Halappanavar and other cases and said that at the very least the Government should commit to have a referendum in the course of this Dáil term.
The Taoiseach said the people voted in the referendum and that was interpreted by the Supreme Court and no government dealt with that until this Government introduced legislation; the Protection of Life During Pregnancy Act.
He said there will not be a referendum on this issue under this Government.
No referendum on voting rights before general election - Deenihan
Minister of State with responsibility for Diaspora Affairs Jimmy Deenihan earlier said he cannot see a referendum taking place before the next general election to extend voting rights in presidential elections to Irish citizens living outside the State.
TDs will give their views in the Dáil later today on broadening voting rights in presidential elections to Irish citizens who live abroad.
It follows a recommendation of the Constitutional Convention in 2013 to do so.
No referendum on voting rights - Deenihan
Speaking on RTÉ's Morning Ireland, Mr Deenihan defended the delay in acting over the issue.
He said that there is plenty of time to change voting legislation before the next presidential election in 2018.
Mr Deenihan said there are around one million active Irish passport holders abroad.
However, he said this number may be greater as many more people may be entitled to an Irish passport.
Speaking on the same programme, Conor O'Neill said extending the vote for presidential elections would be a good stepping stone, but it cannot be the end of the issue.
Mr O'Neill is a recent Irish emigrant living in Brussels and co-founder of the group 'We're Coming Back', which advocates for voting rights for Irish citizens abroad.
He said the efforts of all those who returned home to vote in Friday's referendums highlights that emigrants are keen to engage in matters of State.
He said because of the logistics of organising it, the 2018 presidential election would realistically be the first opportunity.
When asked about emigrant voting rights in the Seanad elections, Mr Deenihan said he would be in favour of it but would not clarify as to whether it would happen in the lifetime of this Government.
