Jeane Freeman told MSPs today that she could not imagine the "pain of losing a child in these circumstances" or their "suffering and grief", and said she wanted "to apologise to them that they feel they have not had their questions answered."
Expressing her "deepest sympathies" to the affected families she said officials were now "urgently seeking full details" of the claims that the child deaths in the Royal Hospital for Children in Glasgow were the result of water contamination.
However, she was criticised by opposition MSPs who said her statement in Holyrood would not have comforted the families affected.
Milly Main, 10, and three-year-old Mason Freeman were patients at the hospital, which is on the Queen Elizabeth University Hospital campus, and died three weeks apart in August 2017. Both were treated in a ward which was later closed because of problems with the water supply,
On Monday, NHS Greater Glasgow and Clyde apologised for the distress caused to parents.
Scottish Labour's health spokesperson Monica Lennon said "heartbroken parents" needed answers and "the truth about water contamination and the multitude of scandals at this hospital must come out."
She added: "Milly Main’s mum deserves the truth about her daughter’s death, as does Victoria Freeman whose son Mason also died unexpectedly at the hospital in 2017.
"Many other families do fear a cover up so I am afraid to say this statement is underwhelming and families deserve much, much better. A passing mention of possible escalation measures against the health board is weak and it’s not good enough.
"It is not clear exactly what the government is prepared to do. I ask the Cabinet Secretary does she have complete confidence in the current leadership? Parents do feel completely failed and the wider public is losing confidence.
"Where is the empathy and compassion for these families? Why should they place their trust in the health board and this Cabinet Secretary?"
But Ms Freeman hit back, and said: "I think they should put their trust in me because I am compassionate I do have empathy and that’s precisely why I met those families and have undertaken the work that I have done.
"I refute absolutely from Ms Lennon or from anyone else that I am careless or irresponsible on these matters. It could not be further from the truth.”
She added: “I don’t accept my reference to escalation is weak - there is a proper process. That process is underway with respect to Greater Glasgow and Clyde. A decision will be taken and I will advise this chamber. I will not be rushed into wrong decisions simply to satisfy members of this chamber.”
Tory MSP Miles Briggs also said the statement would do little to answer the questions of families. "They believe there has been a cover up and we need to get answers," he said.
Asking if all families had been contacted and "provided the duty of care they need", he also questioned what guidance and protocols had been put in place after infections were discovered and what sight ministers had of this.
Ms Freeman said all the families who had been in contact with herself or Professor White had been contacted, and the chair and chief executive of the health board had written to all families whose children had been treated, to offer a meeting.
In her statement, Ms Freeman said the families affected were "absolutely right to ask and pursue their questions, and they are entitled to have them answered and to receive the support they need."
She added: "I take very seriously the concerns highlighted to me about the deeply concerning issues which have been raised by a whistle-blower.
"As you know, the whistle-blower who came forward last week stated that an internal clinician-led review within NHS Greater Glasgow and Clyde (NHSGGC) had identified additional cases of infection amongst paediatric cancer patients – including a child who died in 2017.
"My officials are urgently seeking full details of this review so we can fully understand the findings and what action the board took in response."
Setting out the actions she had taken, she said that following her announcement of a statutory public inquiry on September 17 to examine issues at both the Queen Elizabeth campus and the new, but unopened, Sick Kids Hospital in Edinburgh, she had "received correspondence from a bereaved parent concerning the death of their child in 2017."
Ms Freeman said: "This was the first notification I received about this tragic death. I replied expressing my concern for them and my condolences for their incredibly sad loss. I advised them a representative from the board would make personal contact with them to ensure any questions and concerns were fully addressed.
"I understand that contact has now been made with the family and I am clear that I expect the board to do all that they can to support them and provide clear answers to their questions.
"Over the course of last weekend, other families have made contact with me. It would not be right for me to discuss the details of each case publicly and it would be entirely inappropriate for me to comment further on any case which is subject to an ongoing Crown Office investigation.
"However, I want there to be no doubt that I am fully committed to ensuring that every family receives the answers they are entitled to.
"It is not a requirement – nor would it be appropriate - for the Scottish Government to be notified of every patient death.
"But my clear expectation is that our NHS must support families by providing them with the accurate information they need to understand what has happened, and what is being done. And to do that in a clear, transparent and timely way."
She added that she would confirm the chair of the statutory public inquiry before the end of the year and set out the terms of reference early in 2020.
"In the meantime, the Independent Review led by Doctor Andrew Fraser and Doctor Brian Montgomery is gathering evidence, with a view to publishing its findings in spring 2020," she said.