Following an accident three months ago, Archie suffered “catastrophic” brain damage. He was found unconscious at his home in Southend, Essex, in April.
He is being treated at the Royal London Hospital in Whitechapel, where doctors believe the youngster to be “brain-stem dead”.
On Monday, Court of Appeal judges rejected Hollie Dance and Paul Battersbee’s appeal, after the High Court ruled that doctors could lawfully stop their treatment of the 12-year-old.
They had launched an appeal for the High Court to consider the case for a third time.
Mr Justice Hayden had ruled on July 15 that, after reviewing the evidence of the case at the Family Division of the High Court in the capital that the medical argument for ending life support was “compelling and unanimous”.
He called Archie’s fate a “tragedy of immeasurable dimensions”.
Mr Justice Hayden said: “Archie’s mother described him as a fighter and I have no doubt he was.”
He added: “But the fight, if it can properly be characterised as such, is no longer in Archie’s control.
“The damage to his brain has deprived him of any bodily autonomy.
READ MORE: Macron ‘hurting businesses all over EU’ just to spite Brexit
But lawyers for Archie’s parents took the ruling to the Court of Appeal to argue Mr Justice Hayden had not given “real or proper weight” to the family’s wishes, nor Archie’s known wishes and beliefs.
The Court of Appeal had also been asked to delay the ruling because of Mr Battersbee’s ill health, but they refused this request.
The judges at the Court of Appeal said a judgement delivered on Monday was in Archie’s best interest.
Prior to Mr Justice Hayden’s ruling, Mrs Justice Arbuthnot ruled in June that Archie was dead, and the Royal London Hospital could lawfully end treatment.
Immediately following the June ruling, Ms Dance said: “This is only the start. I will not give up on my son.”
She added: “I feel sickened that the hospital and the judge have failed to take the wishes of the family into consideration.
“I do not believe Archie has been given enough time. From the beginning I have always thought ‘what is the rush?’
“His heart is still beating, he has gripped my hand, and as his mother, and with my gut instinct, I know he is still in there.”