The First Minister was also told she had “broken the law” over the legal duty of candour placed on public institutions when an unintended incident leads to a death.
At First Minister’s Questions on Thursday, Ms Sturgeon was challenged over her government’s handling of the scandal of Glasgow’s Queen Elizabeth University Hospital’s infection, which led to the death of two children.
Scottish Labour leader Anas Sarwar said it was “unforgivable” the mother of one child who died had to “learn the true cause of her daughter’s death in a newspaper and join the dots”, but also for the parents of the second child who died still to not know the truth.
He said the evidence he shared 18 months ago about the death of ten-year-old Milly Main.
Mr Sarwar also said a case note review in March this year had “confirmed over 30 infections in children and two children’s deaths”, yet despite a promise from Ms Sturgeon that all affected families would be contacted, “one family has not been informed and it’s feared this is the family of the second child who tragically died”.
Demanding answers on what attempts had been made to contact the family, he added: "The key question is when that first attempt was made.
"I raised the case in November 2019. The case note review team should not be the ones trying to inform that family, because senior clinicians informed the health board about the deaths of two children months earlier in July 2019.
"Scotland’s duty of candour law means families should be informed as soon as the health board became aware. The family should have been informed 18 months ago, not contacted for the first time just a few weeks ago.
"You have broken that law. To be clear – two children died in Scotland’s flagship hospital, one family had to find out by fighting the health board and another family may not even know.
"This is the biggest scandal of the devolution era. Will the First Minister take personal responsibility that this family will be found and told the truth?”
Ms Sturgeon said she extended her “deepest sympathies” to the families affected “as a result of the issues at Queen Elizabeth Hospital”, but said she could not comment on an individual’s care.
She said: “The expert panel has provided individual reports to families of patients included in the case note review and offered to discuss findings with them. They have contacted all families with the exception of one family. Regrettably and despite extensive efforts by the review team and Greater Glasgow and Clyde NHS, that family has not been able to be contacted.
“I have received reassurance there were rigorous attempts to contact the one remaining family and unfortunately it has not been possible to contact them.
"This is a matter of the most seriousness. The case note review looked at 118 episodes of serious bacterial infection in 84 children. All the families of those children with one exception have been contacted.
“There is one family, that various serious attempts have been made to contact, and it’s not been possible.
"That is deeply regrettable, but it’s not because the health board doesn’t want to contact the family.
"This is about an inability to make contact. I will certainly seek to provide further information as to how many attempts have been made and I will certainly be insuring the health board does everything possible to locate this family.”
The duty of candour legal duty sets out how organisations should tell those affected that an unintended or unexpected incident appears to have caused harm or death.
They are required to apologise and to meaningfully involve them in a review of what happened.
The duty was introduced in April 2018 and applies widely across health and social care organisations, including health boards, hospitals and clinics, GP services, and dental services and surgeries.
Mr Sarwar said Ms Sturgeon had missed the "fundamental point” the case note review “happened as a result of families fighting with the health board to get that review”.
He said: “This scandal involved denials, bullying of clinicians, cover-ups ... clinicians have been raising the alarm for years.
"Inexplicably there are still families fighting for truth and justice. The response from the First Minister has not been good enough. Why has no-one taken responsibility, why have there been no consequences, who is going to be held accountable?”
But Ms Sturgeon said the government had now established a full independent public inquiry to answer the questions Mr Sarwar posed “which is still to do its work and report”.
She said: “He called for a public inquiry. There has been an independent review, a report from the oversight board and the case note review process.
"I dispute that the government is not taking this seriously. We are determined to get to the bottom of everything, from the opening of the hospital to making sure the families have the answers they need.”