In a letter sent to UK counterpart Steve Barclay, Mike Russell sought clarification on Scotland's role after comments from the PM on Wednesday.
Mr Johnson told Ian Blackford the Scottish Parliament had "no role" in passing his Brexit deal.
The SNP's Westminster leader wrote to Commons Speaker John Bercow following the Prime Minister's comments.
Under a political convention in the devolution settlement, consent should be sought from the legislatures of Scotland, Wales and Northern Ireland if the UK Government is looking to rule on devolved matters.
The approval - known as a legislative consent motion - is not legally binding and the UK Government can forge ahead without it.
Mr Russell said a letter was received from Brexit under-secretary James Duddridge on October 20 asking for the Scottish Parliament's consent on the new deal.
He wrote: "During Prime Minister's questions, Mr Johnson said, in response to Ian Blackford: 'The Scottish Parliament has no role in approving this deal. On the contrary, it is up to the Members of this Parliament to approve the deal.'
"The UK Government has already conducted a power grab on the Scottish Parliament when it tore up established constitutional rules during the passage of the EU Withdrawal Act.
"For the first time since devolution was established the UK Government legislated for devolved matters and changed the powers of the Scottish Parliament without the consent of MSPs.
"Now the Prime Minister has gone further and, contrary to the letter from James Duddridge, appears to want to shut the Scottish Parliament out entirely from a process that clearly impacts in many ways on devolved policy areas."
The Scottish Brexit Secretary added: "The idea of the United Kingdom as a partnership of equal nations has been a casualty of the Brexit process.
"The Scottish Parliament that so many people in Scotland voted for must not be cast aside in your Government's pursuit of a Brexit that people in Scotland do not want."
A spokeswoman for the Scottish Government maintained the budget had been set for December 12 but admitted it would be more difficult without a UK Budget first.
She said: "We are concerned that the UK Government is set to delay its Budget, having already scrapped plans for a full three year spending review.
"While we are considering all options available to us, without the Office for Budget Responsibility's tax and economic forecasts that are produced for a UK Budget, we will not have the clarity we require on the funding available for public services in Scotland.
"We have informed the Scottish Parliament of our intention to hold the Scottish budget on December 12.
"We will keep this under review as we monitor events at Westminster."