But lawyers representing the parents Tom Evans, 21, and Kate James, 20, had asked judges at the Court of Appeal in London to reconsider Alfie's case.
The hospital, which won a legal battle to withdraw Alfie's life support, has urged protesters to show respect.
It is thought that Supreme Court justices may now be asked to consider the case.
In further accusations, Alfie's mother was allegedly asked to leave Alfie's bedside after she questioned hospital visiting rules with supporters, known as Alfie's Army, tweeting on Friday: "Alfie's mummy's been told to leave Alfie's side and removed from the ward for asking the nurse is it acceptable for family members now being told they can not see Alfie".
Alfie, born in May of 2016, suffers from an unknown neurological degenerative condition which hospital staff say has reduced him to a vegetative state and corroded his brain matter.
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"We will never give up on you Alfie".
The hearing was conducted in the Court of Justice in Westminster before Lord Justice Davis, Lady Justice King and Lord Justice Moylan.
"Our understanding of the law is that they do, that Alfie's not a ward of the state, that means he's not a prisoner of the hospital, and that the parental rights of Kate and Tom have not been extinguished", Kiska said. He sought to overturn Mr Justice Hayden's decision last week that Alfie's parents may not remove him from Alder Hey hospital.
The court ruled that his life support won't be turned off until the Supreme Court has reached its decision.
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The letter also stated that family members and friends of the parents were last night "shouting over your son Alfie in his cubicle, abusive, filming other patients and visitors, filming staff, and behaving in a manner which was felt to be threatening, intimidating and unsafe by other parents and by our staff".
That hearing came after Alfie's parents failed to reach an agreement with Alder Hey bosses about their son's end of life plan. The judge said the unanimous view of medical experts was that Alfie's brain had been eroded by disease and further assessment was pointless.
Alder Hey said in a statement: "We would ask that noise levels outside the hospital are kept to a minimum and for example auto horns are not sounded".
Chief Inspector Chris Gibson said: "Whilst many people have gathered to protest in a peaceful way, Merseyside Police is now investigating a small number of reports, some of which originate from social media, as well as instances of verbal abuse and acts of intimidation from those outside the hospital".
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