The parents of Archie Battersbee are filing an urgent application to the Supreme Court to hear their case after the Court of Appeal blocked them from taking it to the United Nations.
The Court of Appeal granted a stay of execution until 2pm on Thursday for an application to the European Court of Human Rights (ECHR), but not the UN.
The Christian Legal Centre (CLC), which is representing the 12-year-old's parents, Hollie Dance and Paul Battersbee, said the ECHR has "a track record of rejecting applications from parents in end-of-life cases such as Archie's".
The family wants the Supreme Court to grant permission to appeal the block on an application to the UN Convention on the Rights of People with Disabilities.
The High Court has sided with doctors who want to withdraw Archie's life support against the wishes of his parents. On Tuesday, the Court of Appeal refused permission to appeal the High Court ruling.
Ms Dance said, "It feels wrong that the Court of Appeal have tried to force us down a road which they know will fail and have taken away our rights of taking the case to the UN.
"All we have asked for from the beginning is for Archie to be given more time and for Archie's wishes and ours to be respected. As long as Archie is alive, I will never give up on him, he is too good to give up on.
"When he is to die, we believe it should be in God's way and in God's time. What is the rush? Why is the hospital and the courts so keen to push this through as fast as possible?"
Ms Dance has previously said that prior to the accident in April that left Archie unconscious and brain damaged, he had expressed an interest in Christianity and being baptised. She has also stated that he would not want his life support withdrawn.
The Court of Appeal said that his religious beliefs were "insufficient" to justify the continuation of life support and that continued treatment "would not be lawful".
While doctors have argued that a "planned death" would be more "dignified", Ms Dance says this would be "the most traumatic outcome" for the family.
"Parents need support not pressure. It is exhausting what we have been through. We should not have to endlessly battle the hospital in the courts for what we believe is right for Archie," she said.
"Top judges have told us, however, that this is the law, if this so, the law must change.
"Archie is making medical progression which we have not been allowed to address in court, but we are determined to present it as evidence as take this legal challenge forward.
"We will continue fighting for Archie and will not give up."
CLC chief executive Andrea Williams said, "The pace at which this case is running is troubling and there appears to be an unexplained urgency from the hospital and courts to end life support for Archie."
She called for a "wholesale review of how the system works in this kind of case" so that hospitals "wait until parents are ready rather than bring the full force of the law upon them".
She continued, "Archie's case has focused on two issues of fundamental importance to the sanctity of life and the protection of the vulnerable.
"First, the controversial concept of 'brain death'. Second, the everlasting ethical dispute between those who believe in sanctity of life until its natural end, in God's way, in God's time – and those who believe that 'dignity' requires a choreographed death, be that euthanasia, assisted suicide, or a planned withdrawal of life support.
"We continue to stand and support the family in every way we can as they take their case to forward."