Archie has been left in a comatose state after suffering brain damage.
His mum and dad, Hollie Dance and Paul Battersbee, had asked Supreme Court justices to give them more time to carry on their fight.
Ms Dance and Mr Battersbee want the United Nations to consider the case after losing life-support treatment fights in London courts.
They wanted Supreme Court justices to bar hospital bosses from stopping life-support treatment until they have had time to make an application to the UN.
Paul Battersbee and Hollie Dance
Victoria Jones
But three justices on Thursday refused their application.
A Supreme Court spokeswoman said in a statement: “This afternoon, 28 July 2022, the Supreme Court received an urgent permission to appeal (PTA) application in the case of Archie Battersbee.
“The parents of Archie Battersbee filed their application to appeal to the Supreme Court earlier today. They were seeking a stay of the Court of Appeal’s decision to allow withdrawal of life-support treatment from their child.
“Aware of the urgency of this matter, the court convened a panel of three justices who considered submissions from the parties ‘on paper’, in the usual way.
“Having considered the careful judgment of the Court of Appeal delivered by Sir Andrew MacFarlane (president of the Family Division) and the application for permission to appeal the Court of Appeal’s decision in relation to the stay, the panel has refused permission to appeal to the Supreme Court.”
Family photo of 12-year-old Archie Battersbee
Hollie Dance
Judges have heard that Ms Dance found Archie unconscious on April 7. He has not regained consciousness since.
Doctors treating Archie at the Royal London Hospital in Whitechapel, east London, think he is brain-stem dead and say continued life-support treatment is not in his best interests.
Bosses at the hospital’s governing trust, Barts Health NHS Trust, had asked for decisions on what medical moves were in Archie’s best interests.
A High Court judge, Mrs Justice Arbuthnot, initially considered the case and concluded that Archie was dead.
But Court of Appeal judges upheld a challenge by his parents against decisions taken by Mrs Justice Arbuthnot and said the evidence should be reviewed by a different High Court judge.