Abbott's office has battled falling polls in recent weeks and some within the conservative government's ranks have complained that his advisor Peta Credlin holds too much power.
"Do you really think that my chief-of-staff would be under this kind of criticism if her name was P-E-T-E-R as opposed to P-E-T-A?" Abbott asked state broadcaster ABC.
"I think people need to take a long hard look at themselves with some of these criticisms."
Abbott, who has been criticized for only having one woman in his cabinet- Foreign Minister Julie Bishop, who reports say has a tense relationship with Credlin- said any complaints about his office should come to him.
"This is the same office which ran a very effective opposition, it's the same office which has got an enormous amount done this year sometimes under very difficult circumstances," he said.
Tensions with government MPs emerged this week when Foreign Minister Julie Bishop said Abbott's office had knocked back her first request to attend a UN climate conference in Peru, prompting her to ask again.
Gillard became a torchbearer for women around the world in 2012 with her fiery speech about misogyny on the floor of parliament in which she accused then opposition leader Abbott of sexism.
The unmarried Gillard said she had been offended by many of Abbott's remarks over the years, including that she should "make an honest woman of herself".
Back then, Abbott said it was time "everyone in this parliament moved on from the gender card which so many members of the government have been playing".
Abbott, a fitness fanatic who is also a volunteer surf lifesaver and bush fire fighter, enhanced his macho image when he recently promised to "shirtfront" Russian President Vladimir Putin over the crisis in Ukraine.
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