The army veteran and charity fundraiser – who has died aged 100 after testing positive for COVID-19 – became the target of online abuse for travelling on his "bucket list" trip to Barbados over Christmas – before the new lockdown restrictions came into place.
Morgan revealed on Good Morning Britain that he had spoken to Captain Sir Tom’s daughter Hannah Ingram-Moore and she had admitted how much the comments had hurt the family as they mourned her father.
The GMB host said: “The trolls who have been mocking and abusing Captain Tom and his family – before and after he died – Hannah told me how it hurt them.
“It hurt them at a time when they were at their darkest moment with their dad, their grandad. It just added a little bit more despair and hurt on to what they were suffering as a family.
“And I hope you can live with yourselves, I really do. Because it was despicable. And the very worst of this country was some of the stuff I read on Twitter and social media in the last few days.”
Morgan, 55, called for tougher laws to penalise and prevent social media trolls being able to post and promote online abuse.
Watch: The wonderful life of Captain Sir Tom Moore
He added: “The prime minister rightly condemned it, because we have to do something about this.
“The idea that people think it’s OK to abuse Captain Tom and his family after they raised £39m for the NHS, that they think they can abuse Marcus Rashford and the Black footballers, that they think they can do this with impunity and never suffer any consequences for it and never think about the misery that they inflict on their targets.
“Well, I can tell you you inflicted it on the Moore family in their darkest moment, you really did. And I think that is despicable.”
Captain Sir Tom – who celebrated his 100th birthday last year – had previously spoken of dreaming of travelling to the Caribbean.
After consulting with doctors, he and his family flew to Barbados on 11 December – before Bedford was placed into Tier 3 on 19 December and then Tier 4 on 20 December.
He returned to the UK on 6 January.
Read more: Queen pays tribute to Captain Tom Moore
Between 9 December and 12 January Captain Sir Tom was tested regularly for COVID-19 and each test returned negative.
At the weekend, a message from Hannah on his Twitter account announced he had been admitted to Bedford Hospital after being treated for pneumonia and then testing positive for COVID-19.
Captain Tom’s family confirmed the sad news of his passing on Tuesday 2 February.
In a statement on Tuesday afternoon, Sir Tom’s daughters Lucy Teixeira and Hannah said: “It is with great sadness that we announce the death of our dear father, Captain Sir Tom Moore.
“We are so grateful that we were with him during the last hours of his life; Hannah, Benjie and Georgia by his bedside and Lucy on FaceTime. We spent hours chatting to him, reminiscing about our childhood and our wonderful mother. We shared laughter and tears together.
“The last year of our father’s life was nothing short of remarkable. He was rejuvenated and experienced things he’d only ever dreamed of.”
Watch: The life of Captain Sir Tom Moore remembered