Speaking on the BBC's Good Morning Scotland on Monday, Mr Russell said the general election result shows that the people of Scotland voted "overwhelmingly" for another referendum on independence.
The SNP won 48 of Scotland's 59 seats in Scotland during the vote.
Earlier, Scottish Conservative MSP Michelle Ballantyne said there would likely not be another referendum in the lifetime of the new UK parliament.
Ms Ballantyne also said that the Prime Minister would reject the Section 30 request, which would allow for the devolution of powers needed to hold a referendum.
Mr Russell did not rule out a legal challenge to a rejection of the request - saying "everything is on the table" - however he added it was "not an argument for this week".
Mr Russell said the Prime Minister was taking an "anti-democratic position which, quite frankly, has no chance of succeeding".
"This is bluster from Boris Johnson. This argument took place last week, it shouldn't take place this week," he said.
"Last week, the people of Scotland were asked whether they want the right to choose.
"And they said yes, resoundingly and overwhelmingly.
"The argument is over in that regard."
Mr Russell added: "Are we really going to have a UK Government who holds the people of Scotland in such contempt, that they will not allow them to say what their future should be?"The
Constitution Secretary said the SNP would continue to push for the legal framework for another referendum if the request was rejected.
He added: "We will move forward as a democracy and I hope that Boris Johnson will move forward as a democratic Prime Minister."
On Thursday, the Scottish Parliament will vote on a framework for all referendums in Scotland. Mr Russell said this will give opposition parties the chance to prove they "respect democracy".