The Scottish First Minister argued that it would be a "perversion and subversion of democracy" for her to be denied the right to hold such a ballot after her party's election success.
In Scotland, the SNP increased its share of the vote to 45%, winning a greater proportion of the votes north of the border than the Tories did across the UK, and also increased its number of MPs at Westminster to 47.
In contrast, the Tories - whose campaign in Scotland was focused on opposition to a second independence referendum - lost more than half their Scottish MPs, with this number falling from 13 to six.
But after the Conservatives won a majority across the UK, Mr Johnson telephoned the SNP leader to make "clear how he remained opposed to a second independence referendum".
Ms Sturgeon however claimed that by blocking such a ballot, the Tories were "raging against reality", as she said that "Scotland had chosen a very different kind of future to the one chosen by much of the rest of the UK".
The First Minister, who has already made clear her desire to hold a fresh independence vote in the second half of 2020, said she was going to "pursue the plan I won a mandate for".
Ms Sturgeon, speaking on the BBC's Andrew Marr Show, added: "I think that's the democratic thing to do."
She confirmed that she will "set out this week the detailed, democratic, case for the transfer of power" to Holyrood to allow for a legal referendum to take place.
And she insisted the Tories would not be able to sustain their opposition to such a ballot.
Ms Sturgeon recalled that in the 1980s and 1990s, the Conservatives were "adamant there would never ever be a Scottish Parliament", but added that they "couldn't sustain that position because the will of the Scottish people was too strong and democracy ultimately was honoured".
Speaking about Mr Johnson, she said: "If he thinks - and I said this to him on Friday night on the telephone - that saying no is the end of the matter, he is going to find himself completely and utterly wrong.
"It's a fundamental point of democracy, you cannot hold Scotland in the Union against its will. You cannot just lock us in a cupboard and turn the key and hope that everything goes away.
"If the United Kingdom is to continue, it can only be by consent and if Boris Johnson is confident in the case for the Union he should be confident enough to make that case and allow people to decide."
Ms Sturgeon stated: "Scotland cannot be imprisoned in the United Kingdom against its will.
"The Tories might rage against the reality of what happened on Thursday for a while, I fully expect that they will, but ultimately they are going to have to face up to and confront that reality because of the will of the Scottish people cannot be ignored, that is just the simple fact of the matter."