Seven-month-old Gammy is being raised by 21-year-old Thai woman Pattaramon Chanbua, who agreed to carry a child for an Australian couple for A$16,000 (480,000 baht).
Reports said when the pregnancy developed into twins the couple, who have not been identified, took Gammy's healthy sister but not the baby boy, who also has a life-threatening heart condition.
"It is terrible, just absolutely horrible and heartbreaking," Morrison said of the case.
"But I have got to tell you who is an absolute hero in all of this and that is the Thai mother. She is a saint."
Ms Pattaramon, who reportedly agreed to be the surrogate so she could afford education for her older children and pay back family debts, has spoken of her love for the boy.
But she has also admitted she cannot afford to pay for his medical expenses, prompting an outpouring of global goodwill which has raised more than A$200,000 online for an Australian charity to pay for the infant's care.
"The outpouring of support from Australians I think demonstrates in the best possible way about how I think Australians feel about this," Mr Morrison said, speaking on radio station 2GB.
"We are taking a close look at what can be done here but I wouldn't want to raise any false hopes or expectations.
"Sure there are lots of Australians who are desperate to be parents, but that can never I think sanction what we have just seen here."
Surrogacy Australia president Sam Everingham said the case highlighted the need for better counselling of parents who were intending to engage in overseas surrogacy arrangements.
"You should never be putting in more than one embryo if you are not prepared to take twins," he told AFP.
He also urged couples to use credible surrogacy services overseas, saying it was not uncommon for foreign agencies to embezzle money, go bankrupt or make off with couples' life savings.