Prominent lawyer Joanna Toch chimed in, replying to Burchill with a reference to Markle's mother Doria Ragland.
"No Doria?" she wrote. "Don't black names matter?"
Burchill responded that she "was hoping for Doria Oprah, the racist rotters".
Toch – who was also suspended from her law firm after the exchange went viral, despite apologising and deleting her profile – replied, "Doprah?"
Burchill, however, refused to apologise.
She took to Facebook to announce that she had been sacked by The Telegraph after five years, claiming that she had been unhappy anyway due to the paper rejecting her "edgy column ideas".
In a new essay published in The Daily Mail this week, Burchill reiterated that she was "not upset in the least" about losing her column.
"But generally, for journalism, and for young writers with spirit, it's a very bad thing indeed," she wrote.
"Newspapers with no original voices will decline even more rapidly than they would anyway in the digital age. It's ironic that a conservative newspaper which castigates cancel culture cancelled me for castigating wokery."
Burchill insisted that her tweet was "sober, wry and entirely without racist intent", and that she could not "stress enough how much I deplore the murder of George Floyd".
"What I was mocking was the type of people who – like H&M – live in gated communities while espousing BLM's politics of social upheaval, without giving any thought to the damage that pro-BLM riots do to poor and black Americans," she wrote.
"'Defund the Police', for example – a slogan shrieked by BLM's Marxist leaders in the wake of Floyd's death – causes huge harm to vulnerable minorities, as the orgy of looting and violence in US cities such as Portland, Oregon, has shown."
She also partly attributed the backlash to a "sexist and misogynist element" within "wokeness", "in this instance towards witty women who have a savage streak".
"We live in an age of cultural insanity, a topsy-turvy land where men are women, harassment is justice and the Left are jostling to tug their forelocks and call for those of us who criticise royalty to be punished," she said.
"As Sex Pistol John Lydon put it, 'I never thought I'd see the day when the Right would become the cool ones giving the middle finger to the Establishment and the Left become the snivelling self-righteous ones going around shaming everybody.'"
Burchill touted her move to Substack – an online platform that allows writers to send email newsletters directly to paying subscribers – where she boasts that she "cannot be silenced".
"In the autumn, I'll be back with my book, Welcome To The Woke Trials," she wrote.
"And thanks to Twitter, The Telegraph and a pair of hypocritical fibbers whose fantasy land is so fragile brilliant journalists (me) must be silenced in order to maintain the illusion – it will have a whole new ending."
