It comes after a five-year investigation by a government commission found that 7 percent of Catholic priests allegedly sexually abused children between 1950 and 2010.
The apology would be given later this year, Mr Turnbull said.
The report made 409 recommendations, which Turnbull said his government was now working through. As a nation we will take this opportunity to offer the victims the dignity to which they were entitled as children, and was denied by the people who had to take care of them, the prime minister pointed out.
"We owe it to survivors not to waste this moment and we must continue to be guided by their wishes", Turnbull said.
"Now that those stories have been told, now that they are on the record, we must do everything within our power to honour them", he said.
A similar inquiry is planned in New Zealand, but will only look at abuse in state care, not religious institutions.
Mr Turnbull urged the states, territories and non-government organisations not to delay signing up to the scheme, saying only maximum participation would allow it to be successful.
But some states have been reluctant to sign up to the Commonwealth scheme because they argue there are flaws in the design.
A national redress scheme for survivors is due to begin on July 1, but is contingent on cooperation between state and federal governments.
The government plans to pay about £85,000 to eligible survivors as well as providing counselling and psychological services.
"As of today, not a single dollar has come from any of the states or the institutions whose names and deeds fill the pages of this report", Mr Shorten said.
Leonie Sheedy, of the survivor group Care Leavers Australasia Network (CLAN), gave her cautious support to Turnbull's initiative.
"A national apology to the children, many now adults who endured sexual abuse at the hands of trusted institutions and the people who worked for or with them, is both a simple but deeply meaningful act", said UNICEF Australia CEO, Tony Stuart.
Leaving the NSW Supreme Court where other survivors turned up to mark McClellan's impending retirement, Sheedy told HuffPost Australia that Turnbull must demand churches and other institutions participate in the redress program or lose their charity tax exemptions.