Alfie Evans' parents apologise after reports of intimidation by hospital protesters
Hundreds of supporters gathered outside Alder Hey Children's Hospital in Liverpool, where the 23-month-old boy is on life support.
Merseyside Police said they were investigating after reports of "instances of verbal abuse and acts of intimidation".
Alfie's parents, Tom Evans, 21, and Kate James, 20, want to move their son to a hospital in Rome for treatment.
Mr Evans said in a statement on Facebook: "Deepest apologies from me and Kate James to the parents and staff effected by the protest/demonstration, it wasn't mine Kate's or anyone else intention to harm or cause conflict or upset.
"We just wanted to take our son to give him the chance he deserves."
Chief Inspector Chris Gibson said officers "recognise the sensitivities involved in this very difficult and sad situation".
"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," he added.
"This is extremely unhelpful for all concerned and we are investigating further to establish the full circumstances.
"We would like to remind the public that this is a hospital for sick children and it should not be forgotten that many families are going through extremely challenging and emotional times.
"We would ask protesters to respect families and staff, including the poorly children in the wards and to ensure that access to the hospital is not restricted at any time, so that services including the blood and ambulance service can run as efficiently as possible."
He urged any victims of "intimidation or harassment" to report it to the police or hospital.
Supreme Court justices might be asked to consider the case for a second time after Alfie's parents, who live in Liverpool, used a piece of ancient English common law in their legal battle.
A High Court judge and three Court of Appeal judges have dismissed their claim.
Mr Evans emerged from the hospital two hours after the decision was given by the court in London.
Wiping away tears and visibly shaking, he thanked supporters surrounding him and again asked for his son's life not to be terminated and for other doctors to be allowed to "look after him".
He added: "They can't break us, we are never going to back down.
"Alfie, the family and all of our supporters are stronger than ever and we will keep fighting all the way.
"We will never give up on you Alfie."
Alfie's parents have already lost fights in the High Court, Court of Appeal, Supreme Court and European Court of Human Rights.
But the couple might make another application to the Supreme Court.
Judges have heard that Alfie, born on May 9 2016, is in a semi-vegetative state and has a degenerative neurological condition doctors had not definitively diagnosed.