The former chief of staff to Mr Johnson had produced texts to the BBC appearing to show Mr Johnson dismissing the idea the NHS could be overwhelmed.
The messages said: “I must say I have been slightly rocked by some of the data on Covid fatalities. The median age is 82 – 81 for men, 85 for women. That is above life expectancy.
"So get Covid and live longer. Hardly anyone under 60 goes into hospital (4%) and of those virtually all survive.
“And I no longer buy all this NHS overwhelmed stuff.”
Now the SNP’s Westminster leader has demanded answers, accusing Mr Johnson of pushing to let the virus spread.
Speaking at Prime Minister’s Questions on Wednesday, Mr Blackford said: "Prime Minister, you wrote these words whilst our NHS was facing the darkest moment in its history.
"While doctors and nurses were fighting to contain the pandemic, the PM was actively pushing for the virus to be allowed to run rampant through our towns and cities.
“The Prime Minister was willing, in his own words, to allow the bodies to pile high.
“On October 15th 2020, 60,000 people had already died.”
Mr Johnson accused the SNP Westminster leader of “grossly mischaracterising” what he said, but did not deny it.
He insisted: “I think what everybody in this country understands is that the decisions that we had to take at that time were incredibly difficult.
"And, of course, this in no way detracts from the grief and the suffering of those who have lost loved ones to Covid, whose families have been hit by the consequences of that disease.
“A lockdown also causes immense suffering and loss of life chances and loss of health and to mental health. And in due course he knows very well there will be a chance to look at all of this in a full public inquiry.”
Mr Blackford then demanded the inquiry into the handling of the pandemic start immediately.
He said: “Such a glib attitude towards human life is indefensible. The Prime Minister is simply not fit for office.
“The only way to get to the full truth over this UK Government’s disastrous handling of the pandemic is for this cabal to be made to answer under oath.
“So will the Prime Minister confirm that, in the interests of public health and confidence, the Covid inquiry will begin immediately and commit to appearing at the inquiry himself under oath, before any general election is called?”
The Prime Minister agreed he wanted an inquiry into the pandemic, but insisted it could not happen “in the middle of a third wave”.
Earlier Sir Keir Starmer had highlighted the confusion and changes of policy.
The Labour leader said: “When it comes to creating confusion, the Prime Minister is a super-spreader.
“Why is it OK to go to a nightclub for the next six weeks without proof of a vaccine or test, and then from September it will only be OK to get into a nightclub if you’ve got a vaccine ID card?”
Mr Johnson dismissed his question and accused Sir Keir of trying to “score cheap political points”.