More questions about his leadership were raised after a disastrous state election on the weekend, when the Queensland Liberal-National Party, closely aligned with Abbott's Liberal-National coalition, surrendered the largest political majority in Australia's history after just one term in office.
In a speech dubbed by some commentators as the most important of his political career, Abbott outlined his priorities including job creation, helping families and small business, building roads and strengthening national security.
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He said he had not considered stepping down.
"This will be a test of character," Abbott said in response to a question about his leadership at Canberra's National Press Club.
"Politicians pass the test when they do what is best for the long-term, not when they give in to short-term fear and make a difficult situation worse."
Abbott's tough stance on asylum seekers, his pressure on Russia over the downing of a Malaysian Airlines jet in Ukraine and success on trade deals have won approval but been over-shadowed by anger over proposed cuts to health, education and other services.
Abbott said he believed he had the full support of his deputy party leader, Foreign Minister Julie Bishop, who along with former Liberal Party leader Malcolm Turnbull, has been touted as a potential replacement.
In an effort to shore up support from both his party and the public, Abbott announced he was scrapping his A$5.5 billion ($4.3 billion) paid parental-leave scheme that had been criticized for being too expensive and made without consultation with colleagues.
He said he took responsibility for his decisions.
"I accept that the paid parental leave scheme was a captain's call," Abbott said. "I accept that the restoration of Knighthoods was a captain's call. They are the two captain's calls which I have made but I have listened, I have learned and I have acted."
Haydon Manning, associate professor at Flinders University's school of social and policy studies, said Abbott's cabinet colleagues were unlikely to seek to replace him for now.
"You feel that he will be given through until about mid-year, through the budget process, the selling of the budget, then the litmus test of what the public and party polling are saying," Manning said.
"If it's then clear that Australians have tuned out from this prime minister, then you'd think he would be removed."
A poll on Monday showed approval for Abbott's performance had plumbed fresh lows at 29 percent, versus 67 percent disapproval. Less than a third expected Abbott to lead into the next election, due in about 18 months.