The Dáil has paid tribute to Ruth Morrissey who died of cervical cancer this weekend.
TDs also observed a minute's silence in her honour.
Taoiseach Micheál Martin said the Government acknowledged the failures of CervicalCheck and was profoundly sorry with what was allowed to happen.
He said too many women who should be here were gone because of those failings.
Mr Martin said the Government will implement fully the recommendations of the Scally and McGrath reports.
He said Minister for Health Stephen Donnelly was in the process of appointing a new judge to the CervicalCheck tribunal so it could get started.
The tribunal was promised by the last government and was due to be in place by March. However it has stalled as a result of the Covid-19 pandemic, as well as difficulty in appointing a chair.
Ms Justice Mary Irvine was due to chair it but she has since been appointed president of the High Court.
Mr Martin told the Dáil that he believes a tribunal would be a "far less adversarial than any courts system and that is the experience from a range of historical medical scandals."
Labour leader Alan Kelly told the Dáil that Mrs Morrissey was a national hero who fought the State twice.
He said her biggest legacy is that she enforced the same standard as that of the UK for women in Ireland.
Mr Kelly called for a change to the civil liabilities act so dependents of a dying person could claim for losses rather than having to go through the courts again.
Mr Kelly said that former taoiseach Leo Varadkar had promised that after Vicky Phelan, no woman should have to go through that again.
Mr Kelly said he should never have promised that as he could not deliver it.
He also said Mrs Morrissey should have received a State apology after she had won her case and that today was too late.
The Taoiseach said he committed to working with Labour on changing the legislation and going through it.
Meanwhile, Mr Martin said there would not be a national cervical cancer screening programme if a decision was made to stop outsourcing checks to US laboratories.
Asked if the wrong decision had been made to send samples abroad for testing, he said: "It is not optimal, but equally we simply would not a national cervical screening programme if that wasn't pursued, we need to be honest."
The capacity was not there when the programme started "and is not here now," Mr Martin told the Dáil.
"So we have a decision to make if that is the desired route that people want to pursue, you simply would not have for the foreseeable a national screening programme. I think it saved many lives as well and that has to be said," he said.
Mr Martin was responding to questions from Sinn Féin leader Mary Lou McDonald, who said the outsourcing of screening services was "a scandal."
"I wish to recall that when that decision was made, the House was warned of the many, many dangers of proceeding in that direction. And tragically those warnings have come home to roost," Ms McDonald said.
She said it was essential that the national laboratory that the Government has committed to "is established and resources as a matter of urgency."
Ms McDonald also said it was also essential that when screening is taking place in the US it is invigilated.
Additional reporting Mary Regan