Doctors and nurses have been subjected to abuse in the street and received thousands of threatening messages in recent weeks, hospital chairwoman Mary MacLeod said.
The hospital is in close contact with the Metropolitan Police over the incidents, she added in a statement.
The 11-month-old boy, who suffers from a rare genetic condition and has brain damage, is currently the subject of a legal battle.
Charlie’s parents, Chris Gard and Connie Yates, are campaigning for their son to be allowed to undergo a therapy trial overseen by a specialist in New York. Hospital specialists say the therapy is experimental and will not help. They say life support treatment should stop. The case is due back before a High Court judge tomorrow.
MacLeod said there could 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”.