Video report by ITV News Political Correspondent Libby Wiener
The Commons Leader's comparison came after Dr David Nicholl - a NHS neurologist who helped write Operation Yellowhammer plans for a no-deal Brexit - spoke out over the government’s inability to stockpile medication in that scenario.
His comments on a radio phone-in, in which he suggested people would die in the event of a no-deal Brexit, prompted criticism from the ardent Brexiteer.
Mr Rees-Mogg went on to compare the doctor with Mr Wakefield, who was struck off as a doctor and widely blamed for the scare over the MMR jab.
After an unprecedented intervention by England's Chief Medical Officer Professor Dame Sally Davies, who branded Mr Rees-Mogg's "disgraceful" comments, the leader of the Commons apologised for his remarks.
"I have the utmost respect for all of the country's hardworking medical professionals and the work they do in caring for the people of this country," he said.
"The government is working closely with the NHS, industry and distributors to help ensure the supply of medicine and medical products remains uninterrupted once we leave the EU on October 31, whatever the circumstances."
A Number 10 source said Prime Minister Boris Mr Johnson did not endorse Mr Rees-Mogg's view.
Dame Sally wrote on Thursday that she felt "compelled to express my sincere disappointment in the disrespectful way you spoke to and about Dr David Nicholl".
"Comparing an established medical expert to a man who was struck off the medical register by the General Medical Council, and described by them as 'dishonest, irresponsible and showed callous disregard for the distress and pain of children' is going too far and is frankly unacceptable," she added.
She said that questions over the ramifications of no-deal and "consequent risk to life" are "legitimate".
"Political leaders should be prepared to answer these questions with respect," she added.
Dr Nicholl, who works for Sandwell and West Birmingham Hospitals NHS Trust, previously said Mr Rees-Mogg had “defamed” him.
Dr Nicholl told ITV News: "This was pre-meditated in the House of Commons using Parliamentary privilege.
"I challenge Mr Rees-Mogg to repeat what he said in Parliament outside here or anywhere in public, and I'll sue him."
It's unknown whether Dr Nicholl has accepted Mr Rees-Mogg's apology.
The row began when Dr Nicholl called LBC radio and asked the Commons leader what mortality rate he would accept if the UK were to leave the EU without a deal.
Mr Rees-Mogg told him: “I think this is the worst excess of Project Fear and I’m surprised that a doctor in your position would be fear-mongering in this way on public radio.”
On Thursday in the Commons, Mr Rees-Mogg made the comparison between Dr Nicholl and Mr Wakefield.
“Preparations are in place and they are being done with remarkable efficiency. But yes, a lot of Remainers wish to make our skins crawl,” he said.
“What he had to say, I will repeat it, is as irresponsible as Dr Wakefield in threatening that people will die because we leave the European Union.
"What level of irresponsibility was that?”
On Friday Professor Williams told ITV News she was drawing a line under the row following the MP's apology.