Bosses at the hospital treating seriously ill Alfie Evans have defended staff, as judges rejected a further legal bid to take the toddler abroad.
Protesters have been gathered outside Alder Hey in Liverpool amid a court battle over the 23-month-old's care.
Over the past fortnight, hospital chair Sir David Henshaw said, staff had endured a "barrage" of abuse.
Earlier, the Court of Appeal upheld a ruling preventing him from travelling abroad after life support was stopped.
His parents Tom Evans and Kate James had mounted a last-ditch attempt to allow Alfie to be transported to Rome for treatment.
'Very difficult time'
Lord Justice McFarlane, who led the appeal court panel, said Alfie's parents were trying to take "one last chance".
But, he said, there was no prospect of the couple's challenge succeeding.
In an open letter, Sir David and chief executive Louise Shepherd said staff had been "deeply affected" by Alfie's "desperate" story.
"We share the heartbreak that occurs when a child cannot be cured and when a child dies," they wrote.
"All of us feel deeply for Alfie and his whole family and we continue to do everything we can to support them as best we can, just as we have for the last 17 months".
But, they said, staff had recently been the subject of "unprecedented personal abuse that has been hard to bear".
The hospital has seen several protests in recent weeks, and police have investigated claims patients and staff were intimidated.
Sir David and Ms Shepherd said the previous fortnight had been "a very difficult time."
"As an organisation, we have endured attacks upon our motivation, our professionalism and our ethics," they said.
They described the requirement for a "significant police presence" at the hospital as "completely unacceptable".
Staff had also received "a barrage of highly abusive and threatening language and behaviour that has shocked us all", they said.
"Worse still, patients and visitors to Alder Hey have also reported abuse."
Sir David and Ms Shepherd thanked Merseyside Police for its "unstinting support" and said the hospital's first priority remained "Alfie's comfort, dignity and privacy".
Earlier, Ch Insp Chris Gibson warned social media users the force would act upon reports of malicious communications and threatening behaviour where necessary.
Tom Evans and Kate James wanted to move the 23-month-old from Liverpool's Alder Hey Children's Hospital to Rome.
Lawyers for the couple had urged judges to reconsider the case when Alfie kept breathing despite his life support being removed on Monday.
But the Court of Appeal upheld Mr Justice Hayden's earlier High Court ruling.
At the hearing in London, three senior judges rejected all of the appeal grounds raised by his parents.
Lawyers for Ms James told the court Alfie was "struggling" and in need of "immediate intervention".
Mr Evans' barrister Paul Diamond said if the court had ruled in the family's favour then Alfie would be taken to Italy straight after the hearing.
The couple want to take their son to the Vatican-linked Bambino Gesu Hospital.
Mr Diamond told the court a military air ambulance was on standby "at the request of the Pope".
On Monday, the Italian government granted Alfie citizenship in the hope he would be given an "immediate transfer".
But hours later, his ventilator was switched off and the order preventing him from travelling abroad was put in place.
Alfie, who has a rare undiagnosed degenerative neurological condition, has been in Alder Hey since December 2016.