On April 7, Archie's mother had found him unconscious with a ligature over his head and thinks he might have been taking part in an online challenge.
He never regained consciousness and has since been at the centre of a legal dispute after doctors at the Royal London Hospital said he was "brain stem dead".
Lawyers representing the hospital's governing trust, Barts Health NHS Trust, had told Mr Justice Hayden that Archie had suffered a "devastating" brain injury.
They argued that life-support was "burdensome", "contrary to dignity" and "ethically distressing" for medics treating him.
This is the second time the case has been in the High Court in London.
A different judge previously agreed with doctors and ruled Archie was dead, and now Mr Justice Hayden also ruled in favour of the hospital.
He heard the case in the High Court on Monday after the family went to the Court of Appeal, which ruled a new hearing was necessary.
Mr Justice Hayden said continuing treatment was "futile".
"It serves only to protract his death, whilst being unable to prolong his life," he said.
Mr Hayden said evidence showed that Archie had suffered a "significant injury" to "multiple areas" of his brain and had not "regained awareness at any time".
He said medical evidence had shown that improvement in Archie's condition was "not possible" and there was "no hope at all of recovery".
"Archie's mother described him as a fighter and I have no doubt he was," the judge added.
"But the fight, if it can properly be characterised as such, is no longer in Archie's control."
The judge said he had reached his conclusions with "profound regret".
However, Archie's parents Hollie Dance and Paul Battersbee had argued treatment should continue for as long as his heart was beating.
Ms Dance said the family planned to appeal against the latest decision.
"Archie would want us to keep on fighting... and we will keep on fighting," she said.
Mr Battersbee, who is separated from Ms Dance, told Mr Justice Hayden that his son would "not want to leave" his mother.
He said: “There have been too many battles in too short a space of time", adding that his son "needs more time".
"We'll try to appeal. Who knows?", he said.