Rudd's conservative predecessor, John Howard, set up the Nauru camp under what he called the "Pacific Solution" to process asylum seekers away from mainland Australia.
But the policy was strongly criticised by rights groups in Australia and internationally.
On Tuesday six of the remaining Sri Lankans held at the camp were cleared of rape charges, paving the way for a fresh start in Australia once they pass standard health and character checks.
The Sri Lankans were the only ones still held at the centre after Australia quietly resettled seven ethnic Rohingyas from Myanmar last year.
David Manne, lawyer for 27 of the Sri Lankans, said the dismantling of the Pacific Solution marks the end of an era of "cruelty and injustice to genuine refugees".
The Australian government has not announced a date for the closure of the Nauru centre although a bigger immigration detention facility is nearing completion on Christmas Island, which is Australian territory.
Nauru 'facing collapse'
But the shutdown of the Nauru camp has raised concerns about the future of Nauru itself, with the tiny Pacific island warning it could face economic ruin as a result.
Last week Kieren Keke, Nauru's foreign minister, was quoted in The Australian newspaper urging Australian officials to move cautiously on the closure as it could devastate the island's economy.
He said the country's GDP could be cut by one-fifth, putting the entire economy "at risk of collapsing again".
Nauru's economy recovered from the brink of ruin in 2001 after signing a deal to open the centre generating A$5.5m of the island's total estimated GDP of A$28.5m.
The island also receives an annual A$15m in aid from Australia.
Keke said the government will also be seeking compensation from Australia over the camp's closure.