Hundreds of supporters gathered outside Alder Hey Children's Hospital, in Liverpool, where the 23-month-old boy is on life support yesterday.
Nurses and relatives visiting sick patients were blocked from getting in work as dozens of 'Alfie's Army' supporters spilled into the road outside.
Staff were later forced to call in cops after some demonstrators stormed the hospital's children's paediatric intensive care ward as a mark of protest.
Holding up handmade signs and screaming on megaphones supporters reacted to the news with chants of: "Get him out" and "Alfie is a warrior".
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".
Lord Justice Moylan, who helped reach the court ruling earlier today, expressed his "dismay" at protests outside the hospital.
He said: "We cannot conclude this judgement without recording our dismay or concern of what we have been told of the consequences of what has taken place at the hospital in recent days.
"We are told that some members of hospital staff could not get to the hospital because of road bloackage and that some staff and family members in the hospital could not gain entry."
Earlier top judges ruled to have the terminally-ill tot's life support switched off.
The Court of Appeal agreed with Alder Hey specialists that the terminally ill tot "could not be saved" and that it would be "unkind" and "futile" to continue treatment.
Chief Insp Gibson said: "While 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.
"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.
In a heartfelt plea devastated dad Tom vowed the fight to save his son was "not over".
Taking to Facebook the 21-year-old wrote: "Transferring our stable son MAY be a risk???
"But removing his life support and letting him suffocate and die isn't???????!!!!!!
"Where's the logic in that?
"THEY SAY I NEED TO FACE REALITY!!
"I've been living through it for 15 months.
"Me and Kate accept our son is going to die but when we don't no, so it's only our responsibility to let him outlive his remainder with as much dignity love and defection as possible.
"It's not over!!!!"
Tom and Kate said they have a private jet and ambulance on standby to take their son to Rome.
They have already lost a challenge at the Court of Appeal and failed to have the decision overturned at the Supreme Court and European Court of Human Rights.
Their lawyer Paul Diamond has now said he will apply to the Supreme Court for permission to appeal the decision by 4pm tomorrow.
Alfie, born on May 9, 2016, is in a "semi-vegetative state" and has a degenerative neurological condition doctors have not definitively diagnosed.
Lord Justice Davis, who headed the three-strong panel of appeal judges, told lawyers that at the start of the hearing today that doctors had agreed that there was "no hope".
He said: "We cannot have a kind of legal 'Groundhog Day' where you come back again and again and again on the same point."
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