The former Labor leader, who became Australia's first female prime minister in 2010, also said sexist assumptions about women politicians are everywhere.
Like Clinton, Gillard has been caricatured as scheming, power-hungry and emasculating. That's everyone's problem, Gillard suggested, because women leaders are held to a different standard of likability.
Let's all, Gillard said, just get rid of the old stereotype that “likability and women and leadership don’t go together."
“This is not an Australian question, this is a global question," Gillard said. "We’ll see some of it in real time, I think, during Hillary Clinton’s campaign. And this is the moment for us to be having these deep conversations about how to change it, so that for the daughters in the future there’s none of that baggage.”
Gillard, who left office two years ago, was asked at the Fortune event what guidance she had for Clinton as the American begins her second White House run.
Gillard replied that she had been “absolutely wrong” to assume the brunt of negative reaction to her selection would come at the beginning of her tenure, and then abate. Sexist things kept happening right on through, Gillard said.
"The reaction about gender heightened because it became a sort of convenient cudgel of criticism when you wanted to take a shot at the government," she said.
Claims of sexism among her mostly male colleagues in Parliament led Gillard to make what became her best-known speech, in 2012. The remarks, which quickly became known as the "misogyny speech" came in response to a scandal in Gillard's governing party. Gillard had defended a Labor colleague against calls that he be removed as speaker for sending lewd texts. Then-opposition leader Tony Abbott had accused Gillard of sexism for protecting the man.
"Every day in every way" Abbott himself was sexist and misogynist, Gillard charged.
"I will not be lectured on sexism and misogyny by this man. I will not," Gillard said in her 2012 speech.