Manus Island police say there are no suspicious circumstances surrounding the death of Hamed Shamshiripour.
The 28 year-old was found dead in the jungle near refugee accommodation on the island on Monday.
As anger over his death spreads to Australia, his family has sought legal advice.
They want an inquest into his death conducted in Australia, and an independent autopsy to be done.
The family's lawer, George Newhouse, says they also want Mr Shamshiripour's body to be repatriated to Iran.
"The family want justice. And they want those responsible to be held accountable. When a person dies in custody there is automatic grounds for an inquest in this country and under PNG law."
A letter, published by The Guardian, shows the Australian Border Force Chief Medical Officer, Dr John Brayley, knew about Mr Shamshiripour's mental health problems a year ago.
Mr Newhouse says the refugee had been living on Manus Island since 2013.
"Our leaders are implicated in Hamed's death. Government officials knew about his fragile condition and they left him to die."
Mr Shamshiripour's father spoke to the BBC from his home in Iran.
He told them his son had complained about mistreatment by the Australian government and a lack of action regarding his mental conditions.
A memorial has been held inside the Manus Island detention centre.
It comes amid concerns about another refugee who reportedly fractured his skull and has been transported to Australia in a critical condition.
Iranian-Kurdish refugee and journalist Behrouz Boochani tweeted that the man was beaten and had been vomiting blood for two days.
The UNHCR has warned of an escalating crisis on Manus Island ahead of the forcible closure of the detention facility by October.
Australian and PNG immigration authorities are trying to move refugees to a transit centre in the island's main city, Lorengau, but they have been refusing to go, saying they fear for their safety in an open facility.
The Refugee Action Coalition's Ian Rintoul told a rally in Sydney that refugees and asylum seekers on Manus Island should be brought to Australia.
"Let us say very very clearly from the start: there is only one third country that has the responsibility for the people who have been sent to Manus Island and that's Australia."
The Department of Immigration and Border control has not yet responded to SBS' request for comment.
