His comments come after another dire week for the health service as figures showed more than three quarters of a million people north of the border are waiting on an outpatient appointment, while A&E waiting times hit another record low.
The Health Secretary told the newspaper: “I spend every waking moment trying to resolve the issues of the health service.
“I will leave no stone unturned as long as I am in this role to ensure we get the necessary investment our health and social care systems require to provide you with the standard of care you deserve.
“I am father to two girls who mean everything to me and I would not want them having to wait long periods of time for treatment or diagnosis if it ever came to it.
“I don’t see myself as Humza Yousaf the politician but Humza Yousaf the father, the husband, the son.
“It is important to put myself in others’ shoes to truly empathise with them.”
Mr Yousaf also apologised to those affected by the crisis in the NHS, saying: “If you or a family member have waited for an ambulance too long, waited for treatment for too long, I am sorry.”
He may also have to contend with strike action in the NHS workforce over the coming months, if a new pay offer is not accepted by unions.
Staff including nurses, physiotherapists and ambulance workers are mulling the new deal, which has been described by Mr Yousaf as “best and final”.
According to the Scottish Government, the deal would mean NHS workers in Scotland would remain the best paid in the UK, and workers would get pay rises ranging from £2,205 to £2,751.
For the lowest paid it would be a rise of 11.3%, with an average increase of 7.5%.
Mr Yousaf said strikes would be “really catastrophic” for the health service, adding that other emergency services could be drafted in to help.
But the use of the army, as was seen in recent years to aid the ailing ambulance service, would only be available in the most dire of circumstances.
Elective surgeries and treatments, he added, could be stopped to help ease the pressure during any action.
“We can’t stop people having heart attacks and strokes, the only valve we have is elective care and reducing that or slowing that down,” the Health Secretary said.
“That is something I am very reluctant to do but clearly that is the only real pressure valve we have got.”
Scottish Tory health spokesman Dr Sandesh Gulhane said patients will want “actions to speak louder than words from Humza Yousaf”.
“Saying sorry is merely paying lip service to the ever-deepening NHS crisis unfolding on his watch,” he added.
“He’s a record breaker for all the wrong reasons. The worst-ever A&E waiting times, record numbers languishing on waiting lists, the highest ever number of beds being occupied because of delayed discharge and dwindling GP numbers.”