The US TV appearance saw Markle take part in a prank similar to one used on Ant and Dec's Saturday Night Takeaway, where celebrities are given instructions through an earpiece to carry out embarrassing actions in public.
In his Daily Mail column, Morgan, who quit his Good Morning Britain role earlier this year over a row about his criticism of Markle's Oprah Winfrey interview, wrote that Markle and DeGeneres were "the world's two most narcissistic, fake and irritating female celebrities".
Doing this hidden camera piece with Meghan, The Duchess of Sussex, was one of my favorite things in 19 seasons… pic.twitter.com/rZc2lERzIb
— Ellen DeGeneres (@TheEllenShow) November 18, 2021
Branding the skit "soul-suckingly excruciating", he slammed Markle for embarrassing the Royal Family.
He wrote: "This is a senior member of the British Royal Family whose husband Prince Harry is sixth in line to the throne, behaving like a gormless desperate reality TV starlet.
"Royals don't do this kind of thing for a very good reason."
Arguing that royals needed to keep a level of "mystique", he claimed that the stunt which saw Markle ask street vendors for the hottest sauce they had before telling them "Mommy needs some milk" and drinking from a baby bottle was "puerile and embarrassing".
He added: "For a British royal to demean herself in such a way is completely unacceptable. Not just because it made Meghan Markle look ridiculous, but because it made the British Monarchy look ridiculous."
Morgan again called for the Queen to strip Meghan and Harry of their royal titles and questioned why the Duchess of Sussex would agree to go on The Ellen DeGeneres Show at all.
DeGeneres has been accused of bullying staff on her talk show and has apologised, saying "things happened here that never should have happened".
The interview also included Markle showing off photos of son Archie feeding their chickens and talking about newborn Lilibet's teething problems as she said that she and Harry were very happy.
Watch: Meghan Markle drinks from baby bottle and eats like chipmunk in Ellen prank