Conservative Prime Minister Malcolm Turnbull said he did not feel Mr Rudd was qualified, adding that he would not nominate anyone for the role.
Current Secretary-General Ban Ki-moon steps down at the end of the year.
Mr Rudd has responded by releasing letters in which he says Mr Turnbull had originally promised to back him.
The three letters, published by the Australian Broadcasting Corporation, were written by Mr Rudd to Mr Turnbull between April and July this year.
In one of them, Mr Rudd says that the prime minister had assured him in November and December last year of his backing.
"Based on those assurances, in good faith, I have been informally sounding out governments around the world in terms of their support for my candidature," a letter dated 1 May 2016 says.
"You will appreciate that I would not have been in the business of approaching governments, even informally, had you expressed any doubt about my candidature in any of our previous conversations."
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Earlier on Friday, Mr Turnbull told reporters that his government would not support Mr Rudd's candidacy.
"There is a fundamental threshold point and it is this: Does the government believe, do we believe, do I as prime minister believe that Mr Rudd is well suited for that role? My considered judgment is that he is not," Mr Turnbull told reporters.
He didn't elaborate on why Mr Rudd was not suitable.
A Chinese-speaking former diplomat who also served as Australia's foreign minister, Mr Rudd was a polarising figure within his government, losing the party leadership to Julia Gillard in 2010.
He briefly returned as prime minister in 2013 before Labor lost the general election to the conservative coalition.
He had been considered a long-shot for the position.
Portugal's former Prime Minister Antonio Guterres recently topped the UN's first informal poll of candidates, followed by Slovenia's former President Danilo Turk.
Other top contenders are Argentina's Foreign Minister Susana Malcorra, and former New Zealand Prime Minister Helen Clark.
The Australian government has refused to endorse former Labor Prime Minister Kevin Rudd's bid to become the next United Nations secretary-general.