Under the government's tough immigration policy, asylum-seekers who try to reach Australia by boat are sent to detention camps in the Pacific island nations of Papua New Guinea and Nauru.
They are blocked from being resettled in Australia even if found to be refugees.
The hospital's move came as state governments, churches and activists stepped up their efforts to stop the return of some 267 refugees to Nauru following a High Court ruling.
On Sunday, campaigners from ActionAid, Amnesty International, GetUp! and Greenpeace unfurled a #LetThemStay banner on Sydney's iconic harbour calling for the asylum-seekers, who are set to be deported after being brought to Australia for medical treatment, to be allowed to stay.
The #LetThemStay campaign, which has been trending on Twitter, has also seen hundreds of people maintain a vigil -- now in its third day -- outside the Brisbane hospital where the baby is being cared for.
The 12-month-old infant, who is called Asha and the child of Nepalese asylum-seekers, was brought to the eastern city of Brisbane for treatment in late January after being scalded with hot water at the remote Nauru facility.
Following the High Court's ruling earlier this month in favour of the government's policies, Asha and 36 other babies born in Australia are among the asylum-seekers facing removal.
But a spokesman for Brisbane's Lady Cilento Children's Hospital said Asha "will only be discharged once a suitable home environment is identified".
Their stance was supported by Queensland state's Health Minister Cameron Dick, who said in a statement Sunday that he "strongly support(s) doctors in our hospitals to make the right clinical decisions".
"Doctors must expect to advocate for their patients," Doctors For Refugees co-founder Richard Kidd, who has joined the vigil outside the hospital, told AFP.
"And the situation that we have for children who end up in hospital from Nauru is that we have... overwhelming evidence over many years now that detention does terrible harm to babies and children, particularly their mental health but also physical health."
Australian church leaders in early February vowed to defy the federal government, offering sanctuary to the asylum-seekers.
Several state government premiers have also said they would help settle in their communities those facing deportation if they were allowed to stay.Canberra has long defended its policy, saying it has prevented the deaths of asylum-seekers at sea and secured its borders.