"Put bluntly, we want some urgent action to help these vulnerable people who find themselves in a hopeless, despairing situation," Dr Bartone wrote.
He told SBS News the letter was prompted by a recent groundswell of "concern and agitation" across the AMA membership and the medical profession for the wellbeing of these refugees, especially children.
"A significant number of children are now refusing food and fluids, resulting in severe electrolyte imbalances, which is putting their heart rhythm at risk," he said.
Asked if this could result in deaths of children, Dr Bartone said "when you start seeing electrolytes imbalances causing irregularities in heart rhythm, any possibility needs to be prepared for".
Recent reports have suggested that some children could be suffering from resignation syndrome, a coma-like illness where children withdraw from the world and become catatonic.
Dr Bartone said he could not categorically say if this was the case and that better access should be given to the island to assess these children.
He said the situation had "evolved outside the capacity of the island's medical infrastructure to handle".
"What's happening is entirely unacceptable."
In the letter to the prime minister, he said "the medical situation for the children on Nauru has been described by health experts, including medical staff who have worked on Nauru, as critical and getting worse. It is a humanitarian emergency requiring urgent intervention."
Source: wikimedia public domain
"As a suburban Melbourne GP for more than 30 years, and a grassroots Australian with strong community connections and Christian values, I passionately believe we can and must do more to look after the health of these people, many of whom have fled war, conflict, or persecution.
"The government must get fair dinkum and give these long-suffering asylum seeker children, many of whom are extremely ill, and their families a fair go – bring them to Australia for proper care in the best possible environment for their severe mental and physical health conditions."
Mr Morrison dismissed the letter on Thursday, saying he will not "put at risk any element of Australia's border protection policy".
The prime minister told reporters when a Labor government adjusted the offshore processing policy, "1,200 people died".
He said he was already "getting families off Nauru" through a resettlement deal with the US.
A spokesperson for the Department of Home Affairs told SBS News "Australia provides significant support to the Government of Nauru to provide welfare and health services, including mental health care, education and to support law and order".
"A range of care, welfare and support arrangements are in place to provide for the needs of refugee children and young people. Service providers are contracted to provide age-appropriate health, education, recreational, and cultural services," the spokesperson said.
"The Australian Government has also established a Child and Adolescent Mental Health Service, facilitated through the health service provider in Nauru, International Health and Medical Service."
Australia's offshore policy is designed to deter people from embarking on treacherous sea journeys, with the department recently reiterating that "the Australian Government's position has not changed, these individuals will never come to Australia".