Jacob Rees-Mogg also accused Ms Sturgeon of “waffling on endlessly about having a referendum”.
It comes after Ms Truss labelled the First Minister an “attention-seeker” who should be ignored.
The comments sparked a furious backlash, with Deputy First Minister John Swinney calling them “completely and utterly unacceptable”.
Ms Truss, who spent time in Paisley as a youngster, has described herself as a “child of the Union”.
At Monday night’s leadership hustings she took aim at Ms Sturgeon and her push for another independence referendum in October 2023.
“She’s an attention-seeker, that’s what she is,” Ms Truss said as she rejected the idea of another plebiscite if she becomes prime minister.
“What we need to do is show the people of Scotland, Northern Ireland and Wales what we’re delivering for them and making sure that all of our Government policies apply right across the United Kingdom.”
Brexit opportunities minister Mr Rees-Mogg doubled down on the attack, telling Sky News: “When she’s waffling on endlessly about having a referendum and going to the Supreme Court and all of this, we need to be saying ‘hold on – you’re doing this because you’re failing to deliver for the people of Scotland, and the United Kingdom Government will have to deliver for the people of Scotland as well’.”
He added: “I think she’s very often wrong, she’s always moaning, and we need to focus on how the Union benefits people.”
However, allies of Rishi Sunak, who is also running for the Tory leadership, described Ms Truss’s comments as “unhelpful”.
A campaign spokesman for the former chancellor said: "As members across Scotland will know, it is not good enough to simply ignore the SNP when we need a plan to beat them.
"The SNP Government is letting Scots down every day in running the health service, educating children and tackling crime. Our whole party should be challenging these failings, not ignoring them.
"Winning seats off the SNP in Scotland should be as much of a priority for our next party leader as beating Labour in the next general election.
"Keeping the UK together means confronting the nationalists and beating them at the ballot box. Only Rishi has a plan to do this."
Mr Swinney said “people in Scotland, whatever their politics, will be absolutely horrified by the obnoxious remarks that Liz Truss has made”.
He told BBC’s Good Morning Scotland: “Nicola Sturgeon has far more democratic legitimacy than Liz Truss is going to have if she becomes the prime minister, and I think Liz Truss has absolutely no right or foundation to make these remarks.”
Health secretary Humza Yousaf branded the comments “petty jibes”.
Scottish Greens co-leader Lorna Slater accused Ms Truss of showing “total disrespect” and “contempt for everyone that voted for a pro-independence majority of MSPs in last year’s election”.
She said: “She knows that the democratic case for a referendum is unanswerable, so she would rather patronise us and ignore us.”