The former Scottish Conservative leader said the office of Prime Minister had been "traduced" by Mr Johnson, who was fined for breaking Covid rules alongside his wife, Carrie Johnson, and Chancellor Rishi Sunak.
Lady Davidson, who now sits in the House of Lords, made the comments while out on the campaign trail in Edinburgh with Scottish Conservative leader Douglas Ross, who argues Mr Johnson should not be removed because of the war in Ukraine.
Ms Davidson defended Mr Ross despite disagreeing with him, insisting he had been put in a “terrible position”.
Last night another Conservative peer, David Wolfson, quit as justice minister over the breaches of Covid regulations in Downing Street.
He said in a letter to Mr Johnson that he was resigning not only over the events themselves or the Prime Minister’s “own conduct”, but also “the official response to what took place”.
First Minister Nicola Sturgeon, meanwhile, said using the conflict in Ukraine as a reason why the Prime Minister should stay in office was “the lowest of the low”, and questioned how Mr Ross can “look himself in the mirror”.
Ms Davidson and Mr Ross faced some hostility as they leafleted in Davidson's Mains in Edinburgh ahead of the local elections next month.
Alan Webster, 61, scrunched up material pushed through his letterbox and chucked it away, calling out from his doorstep: “Take this crap away with you. I certainly don’t want it.”
The SNP voter said Mr Johnson should resign, insisting he “always has been a liar”.
Another nearby resident handed their leaflet back, adding: “I don’t trust Tories. Nobody should trust a Tory.”
Speaking to journalists, Ms Davidson said: "For me, to have a Prime Minister who has broken the law, laws that he set, to have presided over a culture in Number 10 where laws were repeatedly broken, and who told both the parliament and the public that no laws were broken when they were, for me that is a resignation offence."
She said Mr Ross had been put in “a terrible position, probably the hardest position that any Scottish leader ever has been", adding: "And as somebody who was the leader for eight years, I know what it's like to have been dumped in it by colleagues down south."
Mr Ross previously led the calls for Mr Johnson to resign, but U-turned following Russia’s invasion of Ukraine.
Ms Davidson said: "I think Douglas was pretty gutsy to come out early, to say that this wasn't on.
"I think that his leadership of the Scottish Conservatives is without doubt, and certainly electorally, the most successful we've ever had - you only need to look at his results in May last year, to find 100,000 more voters across the country than I could ever find is a real achievement.
"But he, like lots of the country, take a worldview wherein there is an international conflict happening right now on Europe's doorstep, and that now is not the time to change prime ministers.
"And I respect that view. It's not a view that I hold.
"For me, when you've got a conflict where it is about supporting a country that is fighting for democracy and freedom, you have to uphold some of the rules of democracy.
"And some of the rules of democracy are that the Prime Minister leads by example, and that he or she follows the rules that they set."
Ms Davidson said that as "a Conservative who believes deeply in the institutions of this country, it's about having the office of Prime Minister being traduced in this way, being held by somebody who has broken the laws that they themselves set".
However, asked if the Tories will be punished by voters in the upcoming local elections, she said: "I don't know. I think that's for individual voters."
The peer added: "I think that there is significant evidence that the Scottish Conservatives repeatedly outperform polls at elections, and I believe we will do so again."
Asked what offended her most about the Prime Minister's actions, Ms Davidson became visibly emotional.
She said: "The thing that really gets me is how many people I know or have met or who have spoken to me, or who I represented when I was still an MSP when Covid was going on, who didn't see loved ones.
"Who stuck by the rules. Who saw folk that they loved die or be ill or be alone, and who will never forgive themselves.
"And they feel that the Prime Minister and the culture in Number 10 makes a mockery of their sacrifice.”
She suggested Mr Sunak should also resign, but stressed: “Who serves in the cabinet is up to only one person and that's up to the Prime Minister.”
Earlier, Mr Ross said the Prime Minister is a truthful man, but stressed he must explain the circumstances around his police fine to the House of Commons.
He said: "I think instability with the change of our leader right now would be the worst possible thing in terms of our efforts against the Russian aggression and it would be the best possible thing for Vladimir Putin.
"But I do not detract in any way from people who take a different point of view."
Mr Ross confirmed the Prime Minister would not be campaigning in Scotland ahead of the local elections.
He said he had discussed his position on Mr Johnson with the Tory MSP group before his statement on Tuesday.
But Tory MSP Brian Whittle later said the Prime Minister should resign, insisting “leadership without accountability is no leadership at all”.
Mr Whittle said: “Boris Johnson should step down and allow a leadership election to take place by the end of the summer so we can put this issue to rest and move on.”
Speaking to Bauer Media, Ms Sturgeon criticised Mr Ross’s position.
She said: "We had Tory MPs trotted out to say that this was no different from teachers and nurses going to the staff room at the end of a shift for a drink.
"I'm sorry, teachers and nurses did not do that at the height of lockdown and it's insulting to suggest that they did.
"And then, perhaps the lowest of the low, the use of the horror and suffering of people in Ukraine as a reason why Boris Johnson should stay in office.
"Douglas Ross has in some ways been leading that particular line and how he can look himself in the mirror in doing so, let alone look anybody in Scotland in the eye, is beyond me.
"At a time of real, serious issues for the UK to confront, surely it's all the more important that the incumbent in Number 10 is a Prime Minister who can lead with moral authority."
The First Minister said the Tories were refusing to abide by long-established "norms of democracy", calling this a "really serious situation".
Scottish Liberal Democrat leader Alex Cole-Hamilton said the Prime Minister’s position was “untenable, and he has to resign for the good of the country”.
Speaking at the launch of his party’s local election manifesto, he criticised Mr Ross, accusing the “lightweight” Scottish Tory leader of having “the backbone of a jellyfish”.