The apology came after senior BMA figures - including top GPs - hit out at the leader of the House of Commons' remarks, aimed at consultant neurologist Dr David Nicholl.
Dr Nicholl, who was asked earlier this year to advise NHS England on preparations for a no-deal Brexit and then spoke out publicly about the risk to patient safety posed by leaving the EU in this way, had clashed earlier in the week with Mr Rees-Mogg on an LBC radio programme.
The consultant neurologist called in during an appearance by the North East Somerset MP on the Nick Ferrari show on Monday, and asked what level of mortality he was prepared to accept as a consequence of a no-deal Brexit.
Mr Rees-Mogg responded during the programme by accusing Dr Nicholl of fearmongering. But on Thursday, speaking in the House of Commons, Mr Rees-Mogg went further with his criticism of Dr Nicholl, in comments that drew an audible gasp from MPs in the chamber.
He said that preparations for leaving the EU without a deal 'have been made', adding: 'A lot of remainers wish to make our skins crawl. I'm afraid it seems to me that Dr David Nicholl is as irresponsible as Dr Wakefield.
'What he had to say in threatening that people will die because we leave the European Union, what level of irresponsibility was that?'
The remarks brought swift condemnation from the medical community. BMA chair Dr Chaand Nagpaul condemned Mr Rees-Mogg's comments Dr Nicholl as 'utterly disgraceful and totally irresponsible'.
The BMA chair said: 'Highly experienced doctors like David Nicholl who decide to speak out about risks to life and patient care, should be supported and listened to, not attacked and derided by those who hold positions of responsibility.
'This unwarranted attack is particularly galling as Mr Rees-Mogg belongs to the same government that called upon Dr Nicholl’s expertise to help draft medical opinion for Operation Yellowhammer and who also wrote the mitigations for the event of a no-deal.'
The former doctor to whom Mr Rees-Mogg compared Dr Nicholl is Andrew Wakefield - who was struck off in 2010 over a now-discredited research paper that claimed there was a link between the MMR vaccine and autism.
Chair of the BMA's representative body Dr Helena McKeown hit out at the comparison on Twitter, calling Mr Rees-Mogg's comments 'dangerous' and asking: 'Who do you trust? A doctor or a politician?'
Dangerous behaviour by @Jacob_Rees_Mogg slandering the highly respected NHS consultant & whistleblower @djnicholl. Our research @TheBMA supports the doctor. Who do you trust? A doctor or a politician? Remember the bus. https://t.co/CFqZ3jXpOt— Helena McKeown (@helenamckeown) September 5, 2019
Following the outcry, in an apology reported by Sky, the Guardian and others, Mr Rees-Mogg said: 'I apologise to Dr Nicholl for the comparison with Dr Wakefield.
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 was 'working closely with the NHS, industry and distributors to help ensure the supply of medicines and medical products remains uninterrupted once we leave the EU on 31 October, whatever the circumstances'.
Health and social care secretary Matt Hancock welcomed the apology - saying he would 'defend to the hilt the right of clinicians and civil servants to provide advice without fear or favour'.