The parents of seriously ill toddler Alfie Evans have launched a further appeal in a bid to take him abroad for treatment.
A High Court judge said the 23-month-old may be allowed home from Alder Hey Children's Hospital, where his life support was withdrawn.
But Mr Justice Hayden refused to allow Tom Evans and Kate James to move their son to a hospital in Rome.
The Court of Appeal judges have retired to consider their decision.
At the start of the hearing, three senior judges heard Alfie, whose life support was removed on Monday, was "struggling".
Jason Coppel QC, representing Ms James, said she had told him her son needed "immediate intervention".
'Clutching at straws'
Paul Diamond, acting for Mr Evans, told judges he had spoken to his client before the hearing, who had told him to "save my boy".
"He would leave no stone unturned... He is clutching at straws," he said.
Mr Diamond argued that there had been a "significant change of circumstances" because Alfie was still breathing after life support treatment stopped on Monday evening.
He said an "alternative" was available, and a military air ambulance was on standby "at the request of the Pope".
"We submit there is a likelihood of Alfie having some pleasure in life," Mr Diamond said.
But appeal judge Lady Justice King disagreed.
She said evidence showed that, while Alfie was unlikely to feel pain, "tragically everything that would allow him to have some appreciation of life, or even the mere touch of his mother, has been destroyed irrevocably".
Michael Mylonas QC, for Alder Hey Children's Hospital, said Mr Diamond had accepted "categorically" before the appeal court and before Mr Justice Hayden that "there is no new medical evidence to contradict the evidence" that was before the High Court in February.
"It was never suggested that death would be instantaneous."
He said Alfie has previously survived six to 10 days without a ventilator and it has "never been said to [his] family that Alfie would die immediately or before sundown".
Mr Mylonas said the "tragedy" for the parents was that Alfie looked like a normal child.
Outside court, two people believed to be German air ambulance staff were escorted from Alder Hey by police.
They were seen speaking to members of the Evans family.
On Tuesday, his parents, backed by the Christian Legal Centre, brought a fresh application for Alfie to be moved from Liverpool to the Vatican-linked Bambino Gesu Hospital in Rome.
At a further hearing in Manchester, Mr Evans claimed his son had fared "significantly better" than expected after life support was withdrawn, suggesting his health had improved.
But Mr Justice Hayden rejected this, saying: "The sad truth is that it is not."
Instead, the judge said, Alfie's continued life was a "shaft of light" and a "special opportunity" for his parents to spend time with him - not the time for more legal manoeuvres.
He also criticised the "malign hand" of one of the family's advisers, law student Pavel Stroilov, who had, the court heard, been party to Mr Evans lodging a private prosecution of Alder Hey Hospital doctors, allegedly for murder.
He said the hospital had provided "world class" care for the child.
The best Alfie's parents could hope for, Mr Justice Hayden said, was to "explore" the options of removing him from intensive care either to a ward, a hospice or his home.
Roger Kiska, from the Christian Legal Centre, told the Victoria Derbyshire programme the judge's comments were "highly inappropriate".
Mr Kiska said the judge was speaking of "a private prosecution of anyone who delivered medicine, drugs, which would hasten Alfie's death, which is de facto euthanasia and a crime in this country".
A doctor, who cannot be named for legal reasons, said for Alfie to be allowed home would require a "sea change" in attitude from the child's family.
He told the court they feared that in the "worst case" they would try to take Alfie abroad.
The toddler has been in Alder Hey since December 2016 with a rare undiagnosed degenerative neurological condition.
Medics have said Alfie's brain has been destroyed by his illness, and it is in his best interests to withdraw life support.
In a statement released after the hearing, Alder Hey Children's Hospital said its "top priority remains in ensuring Alfie receives the care he deserves to ensure his comfort, dignity and privacy are maintained throughout.
"This includes working closely with Kate and Tom as they spend this precious time together with him."
Mr Evans and Ms James had hoped Alfie could be taken to the Bambino Gesu Hospital in Rome, which has links to the Vatican, where his care could continue.
They have lost a series of legal challenges against a High Court ruling in February that Alder Hey could withdraw ventilation.