Togatuki, 23, who had mental health illnesses, died in solitary confinement in Goulburn's Supermax prison.
His sentence for robbery and assault had ended in August, but he was being held ahead of his deportation to New Zealand - the country he had left aged four.
Togatuki had written to Australian Immigration Minister Peter Dutton, begging to be allowed to stay in Australia.
Around 200 New Zealanders are being held in seven detention centres awaiting deportation, after Australia toughened its rules to deport criminals, some who had not lived in New Zealand for many years and committed relatively minor offences.
Some New Zealanders are being held on isolated Christmas Island near Indonesia, and the New Zealand Human Rights commissioner David Rutherford today called for such centres to be closed down.
Mr Key has said New Zealand is unhappy with the situation and he believed Kiwis could be "collateral damage" in a wider policy shift, but today he said it appeared "quite a lot" of those locked-up had serious convictions from sexual offences through to murder.
He wanted to raise the issue in a sit-down with Malcolm Turnbull, but the timing for that would not be immediate as the Australian Prime Minister was not in New York.
"We can register our disappointment there, but doesn't mean we can change it."
Pressure has built for a faster response, and earlier this afternoon Labour leader Andrew Little said an urgent explanation was needed in the case of Togatuki.
"Australians are not treated like this in New Zealand. The Government must do something about this now," Mr Little said.
"It is time our Government stopped sitting back and watching this appalling treatment of New Zealand citizens and sought an urgent clarification of Australia's policy."
Togatuki died after writing farewell messages to his family in his cell, the Sydney Morning Herald reported. Authorities in Australia say he killed himself.
He suffered from schizophrenia and anxiety.
"The death of Junior Togatuki was a preventable tragedy. The Government must seek an urgent explanation about why the 23-year-old - who had a recognised mental illness - was left unsupervised in solitary confinement," Mr Little said.
From last November, Australia has toughened up its rules, and making non-citizens who served more than a year in prison liable for deportation.
That meant there were some people facing deportation who had lived in Australia for many years.
Concerns have been mounting about people, including New Zealanders, held in detention centres, with a member of the Australian Lawyers Alliance saying their treatment could amount to torture.
Today, New Zealand's Humans Rights Commission said Kiwis with concerns about loved ones in Australian immigration centres should contact the Australian Human Rights Commission.
"We support our Australian colleagues' longstanding calls to shut down the Australian Government's detention centres," human rights commissioner David Rutherford said.
"We support this position and encourage our Government to continue to advocate on behalf of New Zealanders held in Australian immigration detention centres."
As well as opposition parties such as Labour and the Green Party, the Government's support partner United Future has condemned the new Australian policy as damaging to the Trans-Tasman relationship.
