The TV presenter, who was one of the judges on Britain's Got Talent when Boyle was discovered, hit out after the 56-year-old was "hounded" by a group of youths near her Scottish home.
The Sunday Mirror claimed as many as 15 youngsters, mainly boys, had been involved in the incidents.
An anonymous witness told the newspaper about one attack which took place on a bus near her home in Blackburn, West Lothian.
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They said: "We were inside the bus and they were throwing stones, screaming and shouting things."
This disgusts me. Vile little vermin
The witness added: Another time 10-15 of them were surrounding and throwing stuff. They lit a piece of paper and threw it at her face.
"Then I was going inside the Mill Centre, Susan was walking out and they were all standing at the entrance and they said to her 'why don't you get yourself a pair of glasses you ugly, old b****. It's horrendous."
Morgan hit out on Twitter, saying: "This disgusts me. Vile little vermin."
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Boyle was diagnosed with Asperger's syndrome in 2012. The syndrome is a form of autism, which is a lifelong developmental disability that affects how a person makes sense of the world, processes information and relates to other people.
Since rising to fame in 2009 on Britain's Got Talent, she has performed for the Pope, the Queen and taken to the stage alongside her idol, Elaine Paige.
Before being diagnosed with Asperger's, her behaviour was thought to be a result of being starved of oxygen at birth.
A Police Scotland spokeswoman said the force had "no comment to make" on the reported incidents.