His parents applied to the United Nations Committee on the Rights of Persons with Disabilities (UNRPD).
The Supreme Court declined to intervene in the case on Thursday.
Hollie Dance and Paul Battersbee, Archie’s parents, argued that the UNRPD has a protocol that allows individuals and families to “make complaints about violations of disabled people’s rights.
Archie has never regained consciousness, and his mother suspects he was participating in an online challenge.
The Supreme Court denied an application to appeal a Court of Appeal ruling that it was legal for treatment to end.
“Words cannot express how devastated we are,” Ms Dance said. The pressure placed on us from the start to hurry through the process of ending Archie’s life was disgraceful. We’ve only ever asked for more time. When other parties have been happy for us to have more time, the hospital and the courts’ urgency is unfathomable.
Archie’s doctors told him he was brain-dead and that continuing life-support treatment was not in his best interests.
Two judges in two separate hearings at the High Court ruled that life support could be legally terminated, prompting Archie’s parents to appeal to the Court of Appeal.
On Monday, three appeal judges upheld the most recent High Court decision.