Mr Russell restated the Scottish Government’s view it has a “cast-iron mandate” to hold a second referendum on independence during a speech in London.
He said he believed Theresa May had refused to permit another vote because she fears it “might be lost by her argument, therefore she does not want to have it”.
Mr Russell’s comments, in an address at the Institute for Government think tank, came just after First Minister Nicola Sturgeon said another vote on independence for Scotland was “as inevitable as it’s possible to be”.
She will make her position on the prospect of a second ballot on the issue known when there is more “clarity” about the UK’s future post Brexit.
Mr Russell said the SNP leader would “give more detail than I can give” on this.
He criticised Mrs May, who has repeatedly said “now is not the time” for Scots to have another vote on independence.
“I find it insulting to be told by anybody, let alone the Prime Minister, that now is not the time and you do not have permission to do something,” Mr Russell said.
“The people of Scotland are mature enough to be able to choose whether we do something or not.”
He added: “I suppose the Prime Minister is basing her argument on the fact that she doesn’t think there is any support for this, I think she should look at opinion polls.
“I think she should look at what the people of Scotland are doing at the ballot box and realise that is no longer the case.
“There is support for this and there is support for independence.
“What I suspect motivates the Prime Minister at the moment is not lack of support for a referendum. It is the fact that the referendum might be lost by her argument, therefore she does not want to have it.”
The Constitutional relations secretary insisted Brexit had highlighted “the very fragile position of Scottish autonomy and decision making”, with Scots having overwhelmingly voted to stay in the European Union in 2016.
While Scots voted by 55 per cent to 45 per cent against independence in 2014, Mr Russell stressed that at the time they were voting for a “United Kingdom that was part of the EU, but also a United Kingdom that was characterised repeatedly during the campaign as a union of equals”.
He said: “Now the UK is set on leaving the EU against the clear will of the people of Scotland.
“That is such an enormous change that it would by itself justify returning to the people of Scotland to reconsider the issue of independence.
“And we did set out these exact circumstances very clearly in our manifesto for the 2016 Scottish general election.”
The SNP won the most seats at Holyrood in that election on a manifesto that set out that the Scottish Parliament “should have the right to hold another referendum ... if there is a significant and material change in the circumstances which prevailed in 2014, such as Scotland being taken out of the EU against our will”, he stated.
Mr Russell said: “No matter what the Prime Minister thinks, we have a cast-iron mandate for a further independence referendum on those grounds alone.”
He said that was a “mandate supported by a majority in the Scottish Parliament, as the Scottish Greens hold the same view, and a majority of Scottish MPs at Westminster”.
Telling the audience he had been “born in that hotbed of Scottish nationalism of Bromley”, Mr Russell said Scotland would continue to have a good relationship with the rest of the UK if it became independent.
He said: “I am in no doubt the social union will continue and underpin whatever eventual relationship Scotland has with the rest of the UK.
“We’re always going to work together, cooperatively and collaboratively. We’re going to have to find ways of discussing matters or mutual interest and concern.”
Mr Russell said that had been the “imperative” for “constructive proposals” the Scottish Government has made to “improving the governance of the United Kingdom” in key areas such as future trading relations and migration.
He said: “The Scottish Government will continue to produce constrictive and considered proposals to improve and enhance the governance of the United Kingdom.”
Mr Russell said Scottish ministers would “shortly” bring forward “new proposals on the reform of devolution”, which would cover statutory foundations for the relationships between the different administrations, dispute avoidance and a new approach to arbitration.
Scottish Conservative chief whip Maurice Golden said: “This is more proof that the SNP is abusing the Brexit process in the hope of engineering another independence referendum.
“Even if senior SNP figures are now taking this message to London, it’s the people of Scotland who are most fed up by it.
“They need to drop the separation agitating and get on with the job voters want them to do.”
Pamela Nash, chief executive of Scotland in Union, said: “Mike Russell should look more closely at the opinion polls. Every single poll for over a year has found that a majority of Scots want to remain in the UK.
“The SNP has a duty to govern for the whole of Scotland, not just the minority who want to leave the UK.
“More than half of people in Scotland do not want a divisive second independence referendum for at least a decade and SNP politicians should start listening to what people are telling them and take this threat off the table.”