The First Minister's challenge follows the SNP's exclusion from Tuesday's televised head-to-head debate between Jeremy Corbyn and Mr Johnson - a format Ms Sturgeon declared was "deeply undemocratic".
A joint legal challenge by the SNP and Liberal Democrats to be included in the ITV debate was rejected by the High Court and, speaking to reporters after the decision, Mr Johnson said he would not debate Ms Sturgeon as she was not in the running to become Prime Minister.
Ms Sturgeon, who is an MSP and is not running to become an MP, argued that a debate between the Labour and Tory leaders does not reflect the choice Scottish voters have at the General Election.
• READ MORE: Boris Johnson: I will never do a TV debate with Nicola Sturgeon"It's deeply undemocratic, in my view, for the SNP not to be represented in that debate," Ms Sturgeon said.
"It's not fair to the SNP, it's not fair to other parties who are excluded, but most importantly it's not fair to the voting public.
"Boris Johnson and Jeremy Corbyn don't properly reflect the choices people in Scotland have - the UK is not a two-party state any more.
"The SNP is the biggest party in Scotland, the third-biggest party at Westminster, we could hold the balance of power after this election.
"Our views should be heard, and indeed scrutinised, in this debate and it's deeply regrettable that won't happen."
Asked about Mr Johnson's comments that he would not debate her, Ms Sturgeon said: "He seems to be a big scaredy-cat.
"He's decided he doesn't want to debate me, you'll have to ask him why that's the case.
• READ MORE: Nicola Sturgeon plans to help Scottish households save on energy bills"I remember when he came to Bute House in the summer and was getting into a debate about independence with me - privately I suggested to him then that we should take it to the public sphere and have a debate in public about that.
"He seemed to be up for that at the time, so I can't really throw any light on what has changed his mind.
"If you want to be Prime Minster - as he is standing in this election to be - then he shouldn't be running scared from debate and I think it speaks volumes that he is so frightened of debating anybody other than Jeremy Corbyn.
"But I'll put down a challenge to him right now: I'll debate him any time, any place. So come on Boris, stop being so scared."
Speaking during a campaign visit to a trampoline activity centre in Stirling with the SNP's candidate Alyn Smith, Ms Sturgeon was also critical of the UK Government's immigration policy, stressing Scotland needs more migrant workers to support its economy.
She said: "There is a need to be frank with people. Scotland's population is such that we need to continue to attract people to come and live and work and contribute to Scotland if we don't want our population to go into decline - leading to less revenue for our NHS and other public services.
"Westminster is taking us on the wrong direction on that - it wants to cut immigration - and that would have a really damaging effect on our economy and public services.
"It's another issue that illustrates the need for Scotland's future and the big decisions that shape our future to be in the hands of the people of Scotland, not in the hands of Westminster politicians who often have the wrong priorities."