Last week the American specialist, Michio Hirano, a professor of neurology at Columbia University Medical Centre in NY, travelled to London to examine Charlie for the first time and discuss the case with Great Ormond Street doctors.
Hospital chairwoman Mary MacLeod said doctors and nurses have been subjected to abuse on the streets and have received thousands of threatening messages.
In a statement, she said numerous messages were menacing, including death threats, and that the hospital was in close contact with the police.
Chris Gard and Connie Yates returned to the High Court ahead of the latest round in their long-running legal battle over their son's treatment.Photo:PA.
The highest courts in the United Kingdom and Europe have sided with the medical professionals at the Great Ormond Street Hospital, repeatedly ruling that continuing treatment offers no benefit to little Charlie Gard.
So far, the courts have sided with Charlie's doctors, who say the treatment is a fruitless endeavor that has not been tested on someone with Charlie's specific condition and that it will only cause the baby more pain.
Pro-lifers in this country and the USA have taken up the case of Charlie while his parents continue their High Court battle, where a judge has been hearing evidence about the potential of the experimental treatment.
Parents visiting their seriously unwell children have also been harassed, including on the grounds of the hospital itself, she said.
"The GOSH community has been subjected to a shocking and disgraceful tide of hostility and disturbance".
The court is expected to rule next week whether or not 11-month old Charlie can be taken to the USA for experimental treatment.
The hospital said police were called after families and staff were harassed. Staff have received abuse both in the street and on line.
President Trump and the Vatican have spoken out in support of Charlie Gard, a terminally ill British infant whose case has captured worldwide attention.
"We fully understand that there is intense public interest, and that emotions run high", she said in Saturday night's statement.
"We recognise the tireless advocacy of Charlie's loving parents and the natural sympathy people feel with his situation".
Some staff have been abused online and others on the street, it said.
On Friday the court heard that a new MRI scan on Charlie made for "sad reading".
Charlie has been examined by Dr. Michio Hirano, an American neurology expert from Columbia Medical Center in NY who has designed the proposed experimental treatment.