The SNP Westminster leader, who last week accused Mr Johnson of being a racist, risked ejection from the Commons for breaching standing orders that prevent MPs from calling each other liars.
However, despite angry Tories attempting to shout Mr Blackford down, Commons Speaker John Bercow made no attempt to discipline the SNP MP.
Mr Blackford called on Theresa May to "admit that neither of the candidates for office should ever be elected prime minister".
Attacking both Mr Johnson and Jeremy Hunt, he claimed the "Tory dream is to drag us out of the European Union no matter what the cost".
Mr Blackford’s comments were brushed aside by Mrs May, who argued Mr Blackford had "voted effectively for no-deal" in refusing to back her Withdrawal Agreement for Brexit.
Mr Blackford said: "Do or die, come what may, those are the words of the Prime Minister's likely successor. The truth behind the Brexit chaos in the Tory party is encompassed in those words. The Tory dream is to drag us out of the European Union no matter what the cost.
"Prime Minister, before you exit office, will you pledge to never vote for a successor willing to impose a devastating no-deal Brexit on all of us?"
Mrs May replied: "As Prime Minister I voted three times in this House to ensure that we could take the UK out the EU with a deal that was good for the whole of the UK, and he voted effectively for no-deal."
Mr Blackford said that was “no answer to a question” and accused the Prime Minister of “gross cowardice”.
He went on: "On the one hand, the Tories are asking people to put their faith in the most incompetent foreign secretary in a century, a man who has made a career out of lying, who has spent his week avoiding the media. Staging photos and playing to the extreme delusions of the Tory shires.
"On the other hand, we have the most incompetent health secretary in our history, a man who writes books on privatising our NHS. The Conservatives clearly don't like the truth."
Mr Blackford added: "In her last days in office, will she finally act in the best interests of these islands - not the Conservative Party - and admit that neither of the candidates for office should ever be elected prime minister?"
Mrs May replied: "Either of the candidates for this high office would do a darn sight better job than anybody sitting on any of those benches."
Later, Tory MP Robert Courts asked Mr Bercow to confirm if the comments had breached Commons rules.
The Speaker said that “if there was an allegation of dishonesty, I didn't hear it”.
"I heard another word used which I don't think was particularly tasteful which I did not judge to be disorderly.
Mr Bercow added: "If there is to be an allegation of dishonesty against a member, that allegation should made on a substantive motion.
"That is the long-established procedure in this House. It should not otherwise be done."