The parents of Alfie Evans are keeping a bedside vigil for the terminally-ill youngster amid tensions between supporters and medical staff.
In posts on Facebook, a family member said Alfie, who is at Alder Hey Children's Hospital in Liverpool, was "doing good", more than two days after he was taken off a ventilator.
Tom Evans and Kate James failed in an 11th-hour attempt to persuade judges to let them move the 23-month-old to a foreign hospital.
The couple, who are both in their early twenties and from Liverpool, maintain that life-support treatment should continue to be provided.
The parents have now lost two rounds of fights in the High Court, Court of Appeal, Supreme Court and European Court of Human Rights.
Alfie's case has touched hearts around the world, with Pope Francis among those who voiced support for the youngster.
However Alder Hey said its staff had experienced "unprecedented personal abuse" from some quarters after it found itself at the centre of a "social media storm".
Hospital workers had been subject to "a barrage of highly abusive and threatening language and behaviour" in person, via phone and online, while there were reports of unauthorised members of the public attempting to enter the hospital.
"Having to carry on our usual day-to-day work in a hospital that has required a significant police presence just to keep our patients, staff and visitors safe is completely unacceptable," the hospital's chairman, Sir David Henshaw, and chief executive Louise Shepherd said in an open letter.
Merseyside Police issued a warning that it was monitoring posts online and that any "malicious communications or threatening behaviour" will be investigated and could lead to action.
Doctors stopped providing life-support treatment late on Monday after Alfie's parents had lost two rounds of fights in the High Court, Court of Appeal, Supreme Court and European Court of Human Rights.
But the couple, who want Alfie to be flown to a Rome hospital, mounted a "one last chance" challenge on Wednesday.
The couple said their son had defied doctors' expectations and his continued survival amounted to a significant change of circumstances which merited a review.
However three Court of Appeal judges dismissed a challenge to a High Court decision made on Tuesday that he should not be taken abroad.
Lawyers representing Alder Hey bosses said Alfie's condition was irreversible and there was no evidence that it had changed.
They said the fact that he had continued to breathe unaided might have surprised members of the public but had not surprised specialists.
Barrister Michael Mylonas QC, who led Alder Hey's legal team, said it had never been suggested that Alfie would die as soon as life-support treatment stopped.
Barrister Sophia Roper, who represents Alfie and takes instructions from a court-appointed guardian, agreed.
Lord Justice McFarlane, who headed the appeal court panel of judges, said Alfie's parents were trying to take "one last chance".
But he said there was no prospect of the couple's challenge succeeding and Alfie was in "the middle" of a palliative care plan.