Campbell, 63, hosted the programme alongside Susanna Reid for three shows and on Wednesday apologised to viewers after accidentally implying the Queen had died when getting her mixed-up with the late Prince Philip.
Realising his mistake, Campbell said: "Can I just say something, I think I may have accidentally announced the death of the Queen. I meant Prince Philip of course."
Morgan, 56, referenced the incident in his Daily Mail diary column, apparently taking delight in Campbell's slip-up.
After stating that he recently had to turn down a hug from presenter Mariella Frostrup due to social distancing rules, he wrote: "I haven’t seen a face so crushed since this morning when Alastair Campbell sealed his brief GMB presenting stint with a grovelling apology for mistakenly telling viewers the Queen had died.
Watch: Alastair Campbell 'accidentally announces' Queen's death
"Ironically, this worst gaffe in breakfast show history may have killed off hapless Al’s scheming aspirations to be the new me. I’d give him a consoling hug, if it was allowed."
Morgan left the breakfast programme earlier this year after refusing to apologise for comment about Meghan Markle's interview with Oprah Winfrey.
The journalist said he did not believe claims made by the Duchess of Sussex, who had made remarks about her mental health and experiences of racism.
Campbell's GMB stint coincided with Mental Health Awareness Week, which he is an ambassador for.
An official replacement for Morgan is yet to be announced by ITV. Since his exit, Adil Ray has assumed most of hosting duties alongside Reid.
Watch: Piers Morgan says he's been thanked by members of the Royal family