The 12-year-old has been in a coma since he was found unconscious by his mother in April and is currently being kept alive by a combination of medical interventions, including ventilation and drug treatments, at the Royal London Hospital in Whitechapel, east London.
His parents, Hollie Dance and Paul Battersbee, were granted a Court of Appeal hearing on Monday after the government asked judges to urgently consider a request from a UN committee to keep treating Archie while it reviews his case.
However, after considering the matter, three judges refused to postpone the withdrawal of life-sustaining treatment beyond midday yesterday.
They also refused to grant permission to appeal against their ruling at the Supreme Court.
Ms Dance and Mr Battersbee filed an application directly with the Supreme Court, asking for his treatment to continue so the UN Committee on the Rights of Persons with Disabilities (UNCRPD) could have time to consider their complaint, made last week.
But, refusing permission to appeal, a panel of three justices concluded the Court of Appeal “made the correct decision”.
In a lengthy statement announcing their decision, Lords Hodge, Kitchin and Stephens said: “As this panel stated in its note of determination last week, the justices have great sympathy with the plight of Archie’s devoted parents who face a circumstance that is every parent’s nightmare – the loss of a much-loved child.”
The judges continued: “It has to be borne in mind that, sadly, the central issue between Archie’s parents on the one hand and the NHS trust, which is supported by Archie’s very experienced guardian, has not been about Archie’s recovery but about the timing and manner of his death.
“As Sir Andrew MacFarlane recorded in his earlier judgment of July 25, there is no prospect of any meaningful recovery.
“Even if life-sustaining treatment were to be maintained, Archie would die in the course of the next few weeks through organ failure and then heart failure.
“The maintenance of the medical regime, as (Mr Justice Hayden) held in his very sympathetic judgment, ‘serves only to protract his death’.
“That conclusion was one which the judge reached only ‘with the most profound regret’.”
Ms Dance said earlier yesterday: “We are having to battle over every decision with the hospital.
“There is nothing dignified in how we are being treated as a family in this situation. We do not understand what the rush is and why all of our wishes are being denied.
“I know Archie’s still with us. Archie’s showing very different signs to what the clinicians are actually putting over to the courts.
“He’s very much there, he’s progressing in so many ways.”