On Monday Court of Appeal judges Lord Justice Davis, Lady Justice King and Lord Justice Moylan ruled against the parents in London.
The parents say that their 23-month-old son is not receiving the care and treatment that he needs.
The apology comes as Alfie's parents today lost their legal battle at the Court of Appeal in London to take their terminally ill son to Italy for further treatment.
Evans has said doctors at Alder Hey children's hospital, in Liverpool, refused to let him remove Alfie from the premises.
The toddler's family have said they are accompanied by their own doctors, have removed duty of care from the hospital and placed it with their air ambulance, and have a jet ready to take the child to Italy.
Alfie's parents were not at the appeal court hearing, which is expected to end in the next few hours.
Court of Appeal judges upheld his decisions.
Last week, Mr Justice Hayden endorsed a detailed plan put forward by Alder Hey doctors for withdrawing life-support treatment, after considering a number of issues at a follow-up High Court hearing.
"Me and Kate need to now focus on spending precious time with Alfie and focus on the next legal step, keeping the relationship with the staff at ease and peace and make sure no parents are being affected any more".More news: Facebook class action looms over facial recognition software
More news: Region told to brace for "potentially historic" ice storm
More news: Bills' Richie Incognito says he's retiring because of health issues
Alfie's parents last week said their son had improved in recent weeks and had asked Mr Justice Hayden to allow a new assessment. The judge said the unanimous view of medical experts was that Alfie's brain had been eroded by disease and further assessment was pointless.
Shortly after the statement, Evans posted a video on Facebook of Alfie's hospital room, from which he said patients could only hear the noise outside if the window was opened.
He also dismissed the suggestion that Alfie was being unlawfully detained an Alder Hey and refused to grant a writ of habeas corpus.
As the hearing began there were more protests outside the hospital by people dubbing themselves "Alfie's Army". "Loud and constant noise, such as from auto horns, affects sleep and raises anxiety levels for our patients, especially when recovering from procedures, so please bear them in mind".
During the December 2017 inspection, safety watchdog Care Quality Commission found "very worrying problems" in its operating theatres, "near misses" that went unreported, and there were complaints from theatre staff that hospital managers did not listen to their "repeated concerns".
Alfie Evans has been battling an un-diagnosed degenerative brain disease.
"As has been determined with considerable clarity in this case, Alfie's best interests are determinative and the court has decided what treatment he should or should not receive". The parents want to move him to a hospital in Rome or Germany.
Mr Evans and Ms James say "the state" is wrongly interfering in their parental choices.