Young Aflie is suffering from an as-yet undiagnosed degenerative brain condition and a judge ruled his life support machine should be switched off.
On Monday, Alfie's parents, Tom Evans, 21, and Kate James, 20, asked Court of Appeal judges to rule that Alfie should be allowed to travel to a hospital in Rome.
The toddler's family have said they are accompanied by their own doctors, have removed duty of care from the hospital and placed it with their air ambulance, and have a jet ready to take the child to Italy.
The latest challenge came as Alder Hey children's hospital employed extra security personnel because of demonstrations in support of the toddler's parents.
"We will never give up on you, Alfie".
However, Alder Hey said in a statement: "Noise from recent protests has unfortunately affected our patients so we would ask that noise levels outside the hospital are kept to a minimum and, for example, vehicle horns are not sounded".
Mr Justice Hayden said flying Alfie to a foreign hospital would be wrong and pointless.
'Alfie's clinical condition is truly heart-rending, but at each stage of the legal process, which has to be followed in such cases, the courts have agreed with the treating team and the independent expert advisers instructed by the trust and the family that Alfie's condition is irreversible and untreatable, ' the hospital said in a statement.More news: Actor Harry Anderson, of 'Night Court,' found dead at Asheville home
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Supreme Court justices and European Court of Human Rights judges refused to intervene.
The letter, from the Liverpool hospital's chief nurse, said the only Alfie's parents and two named visitors were now allowed to visit the Peadiatric Intensive Care Unit (PICO) at one time.
Barrister Paul Diamond, who represented Alfie's parents, on Monday asked the three appeal judges to overturn decisions made by Mr Justice Hayden last week.
Save Alfie Evans! Tell the hospital to let his parents take him home. Moylan argued, however, that both Bambino Gesu staff and staff at Alder Hey agree that no hope remains for curative or preventative treatment for Alfie and that he should receive end of life care.
It has force hospital staff to plead for quiet, fearing noise will affect the recovery of other patients.
"The awful reality was that nearly the entirety of Alfie's brain has been eroded, leaving only water and cerebral spinal fluid", Moylan said, reading from Hayden's previous decision, according to The Sun.
Victoria Georgina tweeted: "Alder Hey are an incredible hospital and as much as i feel for Alfie's parents #imwithalderhey this has turned into something it shouldn't have and the disregard shown to other children, families and the staff who work so hard for our children everyday, is shocking".