Ms Davidson was speaking at the Edinburgh International Book Festival on Tuesday when she told the audience her favourite play was Henry V, by William Shakespeare.
By the end of the Henriad series, Prince Harry has transformed from an irresponsible, wayward youth to a mature and disciplined King of England.
Ms Davidson, who is an English literature graduate, said “no job in the world” can prepare somebody for being prime minister and wishes for a similar change in character for Mr Johnson.
Describing his prospects in office, she said: “I only hope we will see that.”
Ms Davidson acknowledged she has differences of opinion with Mr Johnson – specifically over the dangers of a no-deal Brexit – but called for him to be judged by his actions.
She added: “I’m a strong believer – and was taught in the Territorial Army – you salute the rank, not the person who wears it.
“The country does well if the prime minister does well, so I want him to do well.
“Any background we have is completely irrelevant to the national interest.
“I personally don’t think no-deal is in the national interest – that’s why I’m arguing against it.
“Judge him on what he was like as a prime minister and I will judge on what he’s like as a prime minister.”
The Scottish Conservatives leader predicted the rise in populist ideologies will begin to be rejected by voters, pointing to “Trumpism” in the US and the rise of the right-wing Five Star Movement in Italy as examples.
She went on to say there had “always been a backlash” against “that sort of fundamentalism”, and called for an end to identity politics.
Ms Davidson added: “I think that backlash will come in the UK. You can’t get things done if you believe in absolutism.
“Compromise is such a dirty word at the minute, but it shouldn’t be. We have to get past this identity politics.”
Ms Davidson used Jeremy Corbyn as example of someone who may be dismissed as “just an old Marxist” and not listened to even if he has a good idea.
She added: “The pendulum will swing again – I honestly believe that.”
Ms Davidson was at the festival to speak about her book Yes She Can: Why Women Own The Future, which includes interviews with “world-beating women”.
In the New York Times Theatre event – chaired by Olympic gold medallist Katherine Grainger – she predicted a future where gender inequality was no longer the norm.
She added: “I absolutely see a position in the not too far away future where there’s just no space for that level of misogyny and sexism.”