Meetings were also held this week with Charlie's mother, doctors treating Charlie at Great Ormond Street Hospital and American specialist Dr. Michio Hirano, an American neurology expert from Columbia Medical Center in NY who has designed the experimental treatment.
Pro-lifers in this country and the United States 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.
A judge at the High Court has been hearing evidence about the potential of an experimental treatment that it is claimed could help the baby. Thousands of abusive messages have been sent to doctors and nurses whose life's work is to care for sick children.
Charlie suffers from the rare genetic condition, which causes progressive muscle weakness and brain damage.
Charlie's parents Connie Yates and Chris Gard have fought a high-profile legal battle against GOSH to have their son, who was born on 4 August 2016, flown to America to receive the highly experimental nucleoside treatment.
"Despite conflicting issues, we have always had the utmost respect for all the staff who work tirelessly at Great Ormond Street Hospital and the very hard jobs they do every day".
Hospital chairwoman Mary MacLeod said the London police have been contacted because of numerous threats received by the hospital's employees.
Doctors at GOSH have said there is no realistic prospect of improvement and Charlie should be allowed to die with dignity.
Charlie Gard's case is a heart-breaking one.
"We fully understand that there is intense public interest, and that emotions run high", she said in Saturday night's statement.
Previous courts, including the European Court of Human Rights, have sided with Great Ormond Street Hospital, where Charlie is being treated. Staff have received abuse both in the street and on line.
"We recognise the tireless advocacy of Charlie's loving parents and the natural sympathy people feel with his situation".
'Like them, we have been shocked by some of the public response to this case, and agree with them that it is disgraceful that doctors have received death threats'.
The couple, from Bedfont, west London, say there is new evidence and want Mr Justice Francis to carry out a fresh analysis. The parents said at the hearing it was the first time they were being told about the latest results in the crucial test of Charlie's brain function, according to the AP.
At a pre-court hearing on Friday, a lawyer for the hospital, Katie Gollop, said that the results of the child's latest brain scans make for "sad reading", the Associated Press reported.
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.