Former Liberal Party leader and Communications Minister Malcolm Turnbull asked Abbott to open the party's leadership to an internal vote as the two-year-old conservative coalition government struggles in opinion polls.
The contest will pit a man who has been described as the most socially conservative Australian prime ministers in decades against a challenger some think is not conservative enough.
Unlike Abbott, Turnbull supports gay marriage, wants Australia to replace the British monarch with an Australian president as head of state, and backs a policy of making polluters pay for their carbon gas emissions.
"This country needs strong and stable government and that means avoiding at all costs Labor's revolving-door prime ministership," Abbott told reporters, referring to the opposition Labor Party that changed its prime minister twice in three years.
"The prime ministership of this country is not a prize or a plaything to be demanded. It should be something which is earned by a vote of the Australian people," he added.
Turnbull earlier said the government was doomed to defeat with Abbott at the helm.
"Ultimately, the prime minister has not been capable of providing the economic leadership our nation needs," Turnbull told reporters. "He has not been capable of providing the economic confidence that business needs."
"The disloyalty of some has been outrageous," Hockey told reporters, referring to government lawmakers who have undermined Abbott.
"The prime minister has my absolute loyalty, as I have his," Hockey added.
Turnbull might not be the only candidate. Foreign Minister Julie Bishop, the Liberal Party deputy leader, is regarded as a possible contender, as is Social Services Minister Scott Morrison.
Nick Economou, a Monash University political scientist, was not prepared to predict an outcome.
"The Liberals have done enormous damage to themselves, regardless of the outcome," Economou said. "I find it hard to believe that someone would move on the leadership unless they were absolutely confident of their numbers.
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