Doctors and nurses at London's Great Ormond Street Hospital (GOSH) have been receiving death threats over the treatment of terminally ill Charlie Gard.
The hospital's chairman said the Metropolitan police had been contacted after a series of incidents in which even families visiting sick children have been harassed.
Mary MacLeod said that staff had been subjected to a "shocking and disgraceful tide of hostility and disturbance", with employees receiving abuse in the street and online.
"Thousands of abusive messages have been sent to doctors and nurses whose life's work is to care for sick children," she said.
"Many of these messages are menacing, including death threats. Families have been harassed and discomforted while visiting their children, and we have received complaints of unacceptable behaviour even within the hospital itself."
Eleven-month-old Charlie has an extremely rare genetic disorder, mitochondrial depletion syndrome, that causes progressive muscle weakness and brain damage.
Doctors say he cannot see, hear, move, cry, or swallow, and he has been in hospital since he was 8 weeks old.
A series of courts have agreed with the GOSH decision that Charlie's life support should be withdrawn so he can be allowed to die with dignity.
Charlie's parents, Connie Yates and Chris Gard from west London, want to take him to the US to undergo pioneering treatment. His case has attracted huge international attention, with US president Donald Trump and the Pope intervening.
MacLeod said Charlie's case was 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.
"We recognise the tireless advocacy of Charlie's loving parents and the natural sympathy people feel with his situation."
She added: "Whatever the strong emotions raised by this case, there can be no excuse for patients and families to have their privacy and peace disturbed as they deal with their own often very stressful situations or for dedicated doctors and nurses to suffer this kind of abuse.
"Great Ormond Street Hospital is in close contact with the Metropolitan police and we will do everything possible to hold to account anybody who involved in this kind of deplorable behaviour.”