The woman, "Abyan" (not her real name) is one of two female Somali refugees who have requested help from Australia after being raped on the island. Last week the ABC's 7.30 broadcast distressing footage of the other victim, a 26-year-old woman known as "Namja" (not her real name), calling police for help after her rape - she says it took police four hours to answer her plea.
The 23-year-old woman is 11 or more weeks pregnant as a result of her rape and is begging to be allowed to come to Australia for an abortion. Lawyers acting on her behalf wrote to Mr Turnbull and Immigration Minister Peter Dutton demanding a response by 5pm AEDT Tuesday.
George Newhouse, special counsel at Shine Lawyers, has been making representations to the Commonwealth on the woman's behalf for one week but has had no response.
"Five o'clock has passed, we've had no response from the government," he said.
"I don't know what's wrong with the system but their failure to respond comes across as cruel and heartless.
"Time is quickly running out, this is not a joke, a traumatised and vulnerable woman's health and safety is stake."
Mr Newhouse called on Mr Turnbull to back his condemnation of violence against women with action.
"The Prime Minister wants Australia to be known as a country that respects women, well, we can start with this poor soul," he said.
Asked for comment on Tuesday the Prime Minister's Office referred the matter to the Immigration Minister's office. A spokesman for Mr Dutton said the Immigration Department was aware of the allegations of sexual assault but would not discuss individual cases because of privacy.
"All pregnant women receive professional and coordinated health care," the spokesman said.
"The Department of Immigration and Border Protection is aware of an allegation of sexual assault. Various supports are provided to alleged victims through the government of Nauru Sexual Assault Unit and service providers. Alleged victims are also supported to provide a report to the Nauruan Police Force."
On Tuesday the government of Nauru issued a statement which said: "The government of Nauru says refugees in the country are in no physical danger and stories of locals attacking them are largely fabricated to further political agendas and influence the Australian government."
Mr Turnbull has previously described the rape allegations as "very alarming" and has said Australia is working closely with Nauru to ensure the safety of refugees living on Nauru.
Under Australia's harsh border policies, which are supported by both the Coalition and the Labor Party, asylum seekers who travel to Australia by boat are banished to offshore detention camps on Nauru and Papua New Guinea.
Even if found to be refugees they will never be allowed to resettle in Australia because the Liberal and Labor parties say this would encourage more people to get on boats. The last recorded boat arrival was June 2014, according to Department of Immigration statistics.