"We will never give up on you, Alfie", father Tom Evans, 21, said after the ruling.
Specialists at Alder Hey said life-support treatment should stop and Mr Justice Hayden said he accepted medical evidence which showed that further treatment was futile.
Mr Evans said in a statement on Facebook: "Deepest apologies from me and Kate James to the parents and staff effected by the protest/demonstration, it wasn't mine Kate's or anyone else intention to harm or cause conflict or upset".
Mr Evans said a jet was waiting at John Lennon airport and it is understood he plans to take his son to the Vatican-linked Bambino Gesu Paediatric Hospital in Rome for diagnosis and possible treatment after they received backing from Pope Francis.
"We would ask protesters to respect families and staff, including the poorly children in the wards and to ensure that access to the hospital is not restricted at any time, so that services including the blood and ambulance service can run as efficiently as possible".
In February, Mr Justice Hayden ruled that doctors at Alder Hey could stop treating Alfie against the wishes of his parents following hearings in the Family Division of the High Court in London and Liverpool.
Meanwhile, Mr Evans and Ms James have apologised to families and staff at the hospital following reports of intimidation by protesters.
The court ruled against their case in London yesterday, the BBC reports, and rejected a request for it to go back to the Supreme Court.
Pound surges to highest level since European Union referendum
The pound has also been at its highest level for almost a year against the euro in recent days, at just under €1.16. ET), sterling was marginally lower against the dollar at $1.4313.
Out-of-cash ATMs remind me of demonetisation days: Mamata Banerjee
This panic in Andhra Pradesh and Telangana has resulted in not only heavy cash withdrawals but also people depositing less money. Mr Jaitley today reviewed the currency situation in the country amid reports of its shortage in some parts of the country.
California rejects proposed border duties for troops, USA says
However, previous presidents also saw the need to deploy the National Guard to assist Border Protection agents in the area. It will not be a mission to round up woman and children or detain people escaping violence and seeking a better life.
Barrister Paul Diamond said the couple might make a further appeal to the Supreme Court.
Court of Appeal judges upheld his decisions.
Wiping away tears and visibly shaking, he again pleaded for his son's life not to be terminated, asking that other doctors be allowed to "look after him".
Alfie Evans, who is 23-months-old, has been battling chronic seizures since 2016 as a result of an undiagnosed degenerative brain disease.
"They can't break us - we are never going to back down", he said.
Police have increased security at the children's hospital in Liverpool in fear of angry protests from the parent's supporters, who have dubbed themselves "Alfie's Army".
In a statement, Alder Hey Children's Hospital said it trusted that "the public and supporters of Alfie's parents will read in detail the decision of the Court of Appeal following today's hearing".
According to weekend news reports, Alfie Evans' mother was told to leave his bedside after the hospital informed Alfie's parents that they would no longer be able to sleep in his hospital room overnight.