MPs from the party walked out of the question session in protest, with many shouting "bye!" as they left. But Scotland's devolved parliament, Holyrood, had rejected those measures last month with the backing of all parties save the Scottish Conservatives.
As the noise level began to subside, Mr Bercow said it was a pity as Nationalist MPs had questions down on the Order Paper.
As the chamber descended into confusion, Mr Bercow hurriedly consulted the Commons lawyers and suggested a vote on Mr Blackford's request be taken after PMQs.
Pointing to how there was just 15 minutes on Tuesday night for debate on the Scottish clause to the EU Withdrawal Bill, the PM blamed Labour for using "procedural manoeuvres" in calling unnecessary votes. That is a democratic outrage.
He called on her to bring forward emergency legislation so that the "will of the Scottish Parliament can be heard and, more importantly be respected". The Scottish Parliament had voted overwhelmingly to deny consent for what is a naked power grab.
"That is not acceptable and the Speaker refused to allow a division which I rightly called for".
What an absolute disgrace.
"Our interests have been ignored, the voice and views of the Scottish Parliament have been completely cast aside, and I don't think it will be quickly or easily forgotten". "I will do that".
Turning to the speaker, he said: "Under the circumstances, given the disrespect shown, I have got no option but to ask that this House sits in private". It is not acceptable.
"I think the people of Scotland have to reflect on what has happened here - a Conservative government has stripped powers from the Scottish Parliament and is not prepared to give us the opportunity to debate this". It leaves Holyrood exposed to having powers taken without consent in areas including trade deals, environmental standards and food safety, all issues of key concern to the Scottish parliament and all of which impact detrimentally on Scotland's economy - we're talking a loss of up to 80,000 jobs and cuts to wages of £2,000 a year, according to economists.
Scottish Secretary David Mundell said the SNP was about to be granted a debate on the devolution aspects of the EU Withdrawal Bill, something which did not take place as a result of the walkout. He added: "Disappointed, if not surprised, that if they really felt so strongly about it, they chose a stunt over holding the Government to account".
Ian Blackford was thrown out of the chamber by the Commons Speaker because he refused to sit down after demanding a new debate on Scotland and Brexit.
Meanwhile in the chamber, Mr Bercow at the end of PMqs looked up and praised a baby in the public gallery, who, he said, had been "a model of impeccable behaviour" despite the "turbulence" of the proceedings.