The Scottish Labour leader was speaking out in the aftermath of a row between politicians either side of the border in response to comments made by the foreign secretary about Ms Sturgeon.
The Tory leadership hopeful branded Nicola Sturgeon an “attention seeker” who is “best ignored” at a hustings event in Exeter on Monday, prompting claims that Scottish voters were being undermined.
Mr Sarwar said on Wednesday that while he does not think Ms Sturgeon should be ignored, he does want to see her held to account – and suggested that both figures were “throwing red meat to their own bases”.
“The first thing I would say is Nicola Sturgeon is not Scotland,” he said.
“So all of this rhetoric around ‘this is about ignoring Scotland’ and all the rest of it, I think we should be guarded about that.
"Nicola Sturgeon is not Scotland, the SNP is not Scotland.
“Nicola Sturgeon is a politician like the rest of us, and if the rest of us can be criticised by other politicians and by the media, so can Nicola Sturgeon. She can be criticised by other politicians and the media.
“But in saying that, I don’t want Nicola Sturgeon to be ignored. I want her to be held to account.
“I don't want Liz Truss and [former chancellor] Rishi Sunak to be ignored. I want them held to account.
“I want them challenged, their failures exposed, and for us to have a different government, both across the UK and her in Scotland.”
Mr Sarwar added: “I think there’s a lot of attention-seeking happening in Scottish and UK politics, and so I’m not going to criticise anyone for seeking attention, because I seek my own attention, as you can imagine.
“But what I think is the important point here is, I don’t want Nicola Sturgeon to be ignored.
“I want her to be held to account.”
The Scottish Labour leader said both Ms Truss and Mr Sunak, who are competing to take over as Tory leader and prime minister from Boris Johnson, were “a gift to the Labour Party”.
“They’re both continuity Boris Johnson,” he said. “They’re both candidates who are just Boris Johnson with more combed hair and and probably less of the jokes.”
But he warned that focus should be put towards ongoing issues affecting the public, including the cost-of-living crisis and problems within the NHS.
“We currently have two governments who play on conflict, who want to run government based on conflict rather than co-operation when it’s in the national interest,” he said.
“The real issue that gets ignored is what’s happening to people’s lives, and right now we’re in the midst of a cost-of-living crisis, the biggest fall in living standards since the war.
“I want our governments focused on that. Record numbers of people are on the NHS waiting lists, 700,000 people in Scotland are on NHS waiting lists. I don’t want that ignored.”
SNP MSP Paul McLennan said: “Anas Sarwar had the opportunity to call out the likely next Tory prime minister for this clear contempt for Scotland and the democratically elected First Minister – but he didn’t take it.
“It’s no wonder Scottish Labour has absolutely zero credibility in Scotland. They choose to stand shoulder to shoulder with the Tories in their Trump-like denial of a democracy in Scotland, despite the cast-iron mandate given in a democratic election.”