Mr Rudd made a last-minute dash to Sydney this morning in the hope of lobbying Mr Turnbull to nominate him as secretary-general of the United Nations.
But on arrival the Prime Minister called him and said the Government would not be backing him.
On Friday evening a spokeswoman for Mr Rudd released three letters which Mr Rudd had sent to Mr Turnbull about the issue.
In one, dated May 1, 2016, Mr Rudd said he was shocked to learn Mr Turnbull would not be backing him, claiming he had expressed support as recently as December.
"You will recall that last September I contacted you asking for guidance on how I should address the matter of your previously stated support to me for my candidature when I met Foreign Minister Bishop at the UN General Assembly in September," he wrote.
"You in fact sent me a message on your preferred Wickr system where you stated that you and the FM were 'as one' in your support for my candidature.
"…We continued this discussion further on Wednesday 23 December in your Sydney office. Once again you stated your support for my candidature. You added that when the time came to lodge my nomination, you now wanted to take it to Cabinet to avoid the perception of a 'captain's pick'. You also said to me that the Cabinet process would not change the outcome.
".... You will understand therefore how shocked I was to receive your telephone call within the last couple of hours, just prior to your taking the matter to Cabinet in Canberra.
"In your telephone call you said that neither you nor the Cabinet would be supporting my nomination.
"When I asked the reasons for this, you said that neither you nor the Cabinet has the view that 'I had the qualifications for the position.'
"You will appreciate that you have never expressed that view to me in the multiple conversations we have had on this matter on the past."
Letters released 'in interests of transparency'
Mr Rudd's spokeswoman said Mr Rudd had decided to release the letters because he believed Mr Turnbull was briefing reporters on his version of events.
"As you are aware, Mr Rudd had a conversation with Mr Turnbull this morning about Mr Rudd's interest in becoming a candidate for UNSG," she said.
"Mr Turnbull said he would not be supporting Mr Rudd for that position.
"He also asked Mr Rudd to keep the confidentiality of their conversations on this matter.
"Mr Rudd has unfortunately now discovered that briefings are occurring conveying the Prime Minister's version of events.
"In the interests of transparency Mr Rudd thinks it is far better that he release the three letters he sent Mr Turnbull on this issue over the past few months."
Turnbull said Rudd 'not suited' for job
Speaking in Sydney earlier, Mr Turnbull told reporters he had decided the former prime minister was not suitable for the role.
He said the Federal Government would not be nominating anyone for the role.
"When the Australian Government nominates a person for a job, particularly an international job like this, the threshold question is, 'do we believe the person, the nominee, the would-be nominee is well suited for that position?'" he asked.
"My judgement is that Mr Rudd is not, and I've explained to him the reasons why."
Mr Turnbull refused to be drawn on the reasons why he thought Mr Rudd was unsuitable for the role, or on whether his decision was about satisfying certain members of his party.
Foreign Minister Julie Bishop had pushed Mr Rudd's case during a Cabinet meeting on Thursday, with support from Attorney-General George Brandis.
After the meeting Mr Turnbull said he alone would make a decision on the issue, after Cabinet failed to reach a clear decision.
Decision shows Turnbull is weak, Labor says
Acting Opposition Leader Tanya Plibersek slammed the decision, which she said undermined the authority of both Mr Turnbull and Ms Bishop.
Speaking in Townsville, Ms Plibersek said "the right-wing extremists in the Liberal Party have got the Prime Minister on a short leash".
"He was not able to insist that what any right-thinking Australian would think, that an Australian should be backed for an international job like this," she said.
"He was not able to get that through his Cabinet. It shows the weakness of the Prime Minister, it shows the weakness of the Foreign Minister."
Labor frontbencher Anthony Albanese tweeted: "Turnbull. Pathetic".
Mr Rudd had previously received vocal support from former Liberal leader Brendan Nelson, though South Australian Liberal senator Cory Bernardi had urged his colleagues not to endorse a "dysfunctional", "vengeful", "unstable", "megalomaniac" like Mr Rudd.
Senator Bernardi released a statement after Mr Turnbull made his announcement, congratulating the Prime Minister on his decision and saying: "Our participation in international institutions is more important than an individual's ambition".