Fianna Fáil say the State's investment must be protected
The opposition say there needs to be clarity on the charitable status of the new National Maternity Hospital.
It comes after the Religious Sisters of Charity announced it would have no involvement in the facility by giving up ownership of the St Vincent's Healthcare Group (SVHG).
It follows months of outcry over ownership of the hospital, which is to be built on a Dublin site owned by the healthcare group.
The Religious Sisters of Charity are majority stakeholders in that group.
The congregation said: "The Religious Sisters of Charity will have no involvement in this new company."
"It is now time for us to relinquish completely our involvement in SVHG."
The healthcare group said ownership of the group is to be transferred to a newly-formed company with charitable status simply to be called 'St Vincent's'.
The land the hospital is to be built on, which is currently subject to a rental agreement, is to be sold to SVHG at "commercial terms to be agreed, and the rental agreement will cease", it adds.
Fianna Fáil health spokesperson, Deputy Billy Kelleher, has welcomed the move.
"This decision, I believe, is in the best interests of both the organisation and of patients, and will provide a blueprint for other religious organisations to follow.
"This decision will allow the (health) minister develop a new governance model that ensures clinical independence and autonomy for the new National Maternity Hospital.
But he says the new charitable status will have to be scrutinised.
"Minister Harris must now come forward with a proposal that ensures that the State's €300m investment in the new Maternity Hospital is reflected, and protected, in its ownership and governance structure.
"Compliance with both national and international best practice, and adherence to the laws of Ireland should be the basic benchmark for any hospital providing care to the people of Ireland."
"We don't want to put obstacles in front of this move - but at the same time we do need to have certainty around the State's investment, the recognition of the State's investment and that down the road if there's any other changes that the State has a say in terms of the National Maternity Hospital and what happens to it.
"And that's part of a broader conversation that's required with regard to the State investing in health facilities that aren't presently owned by the State itself", he added.
Sinn Féin spokesperson on health Louise O'Reilly said while she welcomed the Sisters of Charity decision, questions remain.
"We cannot stand over a situation whereby the State gives away a €300m State asset to a newly-formed company with 'charitable status'.
"Such a situation is absolutely ludicrous.
"While the good news that there will be no religious interference in this hospital may obfuscate what has really happened here, I for one will continue to demand that the National Maternity Hospital stays completely in State ownership."
"The fight will go on to ensure that our new National Maternity Hospital stays completely in state control," she said.