At Prime Minister's Questions, Mr Blackford asked Theresa May if she agreed with Mr Johnson "that the Scottish people are a verminous race that should be placed in ghettoes and exterminated".
The SNP MP was referring to a poem by James Michie, published in the Spectator in 2004 when it was edited by Mr Johnson, then the MP for Henley.
Mrs May said the Conservatives "welcome the contribution of people from every part of this United Kingdom, because that's what makes the United Kingdom the great country that it is, and long may Scotland remain part of it."
Mr Blackford followed up by saying: "Words matter and actions matter. The man who published those words in his magazine, the Prime Minister thought was fit for the office of her top diplomat. And he hasn't stopped there - he said that Scots should be banned from being Prime Minister... and that a pound spent in Croydon was worth more than a pound spent in Strathclyde.
"This is a man who is not fit for office... does the Prime Minister not realise that not only is the member racist, he is stoking division in communities and has a record of dishonesty."
Mr Blackford's comments provoked outrage on the Conservative benches, with shouts of "withdraw" from Tory MPs.
Speaker John Bercow intervened to warn the SNP MP to be "extremely careful in the language he uses" and said it would be "much better for him to withdraw any allegation of racism; I don't think this is the right forum, I don't think that is the right way to behave".
Blackford said he obeyed Commons convention by informing Mr Johnson of his remarks before making them, adding: "The member has called Muslim women letterboxes, described African people as having watermelon smiles, and [used] another disgusting slur that I would never dignify by repeating. If that's not racist, Mr Speaker, I don't know what is.
"Does the Prime Minister honestly believe that this man is fit for the office of Prime Minister?"
Mrs May told Mr Blackford that "the purpose of Prime Minister's Questions is to ask the Prime Minister about the actions of the government... I believe any Conservative Prime Minister in the future will be better for Scotland than the Scottish National Party."