Tharnicaa Murugappan was taken to a Perth hospital after being unwell with a fever, vomiting and dizziness.
Advocates say she has since been diagnosed with sepsis and pneumonia, and is in a stable condition.
But they criticised the delay in obtaining adequate medical care.
Tharnicaa had been sick for about 10 days before authorities took her to the hospital on Christmas Island, and later transferred her to Perth.
Her mother, Priya Nadesalingdram, said she had repeatedly asked for antibiotics or hospital treatment, but doctors only gave her paracetamol and ibuprofen pills.
Australian authorities have defended the level of care given to the family on the island, telling the BBC that Tharnicaa had received treatment on Christmas Island "consistent with medical advice".
"The Australian Border Force strongly denies any allegations of inaction or mistreatment of individuals in its care," a spokesperson said in a statement.