The Tories lost the Wakefield seat in West Yorkshire to Labour and Tiverton and Honiton in Devon to the Liberal Democrats – the latter by a swing of nearly 30% – on Thursday.
The results come just weeks after a confidence vote saw 148 of Boris Johnson’s MPs back his removal – including four Scottish Conservative MPs.
Following the by-election defeats, Conservative Party chairman Oliver Dowden quit and opposition parties sought to pile on the pressure.
Speaking from the Royal Highland Show on Friday, Scottish First Minister Ms Sturgeon urged Tory MPs to remove the Prime Minister.
“It’s a monumental, massive, humiliating vote of no confidence in Boris Johnson,” she said.
“People could see it coming 100 miles off and it should send a very strong message – if not to Boris Johnson, who is impervious to all of these things given his arrogance, it should send a very strong message to the Conservatives.
“If they keep Boris Johnson, a law breaking Prime Minister who has been seen to not tell the truth, if they keep him in office, then effectively they are all becoming complicit in this.
“If this is not a wake-up call to the Conservative Party, then it will just prove beyond doubt that they don’t have the interests of any part of the UK in mind or at heart.”
Meanwhile, Scottish Tory MP Andrew Bowie said his party can no longer act as though it is “business as usual” following the losses.
The West Aberdeenshire and Kincardine MP said Thursday had been a “difficult night” for the party, echoing the sentiments of Mr Dowden.
Speaking to BBC Radio Scotland’s Good Morning Scotland programme on Friday, Mr Bowie said he would vote against the Prime Minister again if presented with a choice – but he stressed the country needs to be focused on the “big issues”.
He said: “I made my feelings quite clear only two weeks ago in the vote of confidence when I voted that I had no confidence in the current leadership of the Conservative Party.
“I do believe that we shouldn’t be navel-gazing and worrying about who is the current leader and we do need to be focusing on the big issues facing the country, such as the cost-of-living situation and supporting the government of Ukraine.
“But no, I don’t think we can carry on as if it’s business as usual.”
When asked what MPs should do about their lack of faith in Mr Johnson, Mr Bowie said: “I think that is for others to be discussing this morning.”
However, he said voters had made their opinions clear at the by-elections and many MPs and party members will now be “discussing amongst ourselves how we move forward”.
He added: “I’ve been in this game long enough to know that predicting anything is a fool’s game.
“I wouldn’t want to hazard a guess what is going to happen in the next hour, let alone the next day or the next week or month.”
Scottish Labour leader Anas Sarwar said the people of Wakefield and Tiverton and Honiton “recognise that they deserve better” and challenged his party to set its sights on the next general election and a future Labour government.
Mr Sarwar also told LBC that the SNP’s “last big argument” for independence was Boris Johnson and the Tories.
He added: “I think if we can demonstrate to people that you can express your anger for the Tories, you can get the change you want in our country to bring democracy closer to home, but also bring economic and social powers closer to home and change closer to home without going for independence in a referendum by booting out the Tories in the next general election, that’s a huge shift in mindset in Scotland.”