Mr Morrison's comments came as funds raised online by an Australian charity to pay for the infant's medical care rose above $180,000 after a flood of international goodwill for the infant's plight spurred donations from more than 4,500 people.
The boy, Gammy, and a twin sister were born to Thai woman Pattaramon Chanbua in December after she was reportedly paid A$16,000 (476,800 baht) to be a surrogate.
An unnamed Australian couple took the sister, who was healthy, but left Gammy behind, according to media reports.
The baby boy also suffers from a life-threatening heart condition, and 21-year-old Pattaramon had earlier said she could not afford to pay for the medical treatment he needs.
"I think this is an absolutely heart-breaking story, it really is," Mr Morrison told the Australian Broadcasting Corporation.
"I think perhaps this may fall more into the territory of what people's moral responsibilities are here.
"I note there was a comment earlier that the mother of this child, baby Gammy, wants the child to remain in Thailand and that mother's wishes also have to be absolutely respected."
Ms Pattaramon told Fairfax Media late Saturday she wanted to take care of the boy in Thailand, saying: "I'll take care of Gammy on my own. I'll not give my baby to anybody.
"I wish they will love my baby ... I forgive them for everything. That is the best thing I can do, forgive ... it is best for everybody," she said of the Australian couple.
"I want to see all my children back together again. I don't really think too much about the Australian couple. I can't blame them ... I don't feel upset or angry about them anymore. They might have their own problems too."
Mr Morrison said the surrogacy case had some "serious issues" that needed to be managed very carefully.
"But this whole issue I think is fraught with all of these difficulties and I can understand the longing and anguish of parents in this situation who want to be parents, but equally there are some serious issues here that have to be managed very carefully," he said.
Fairfax Media earlier reported that a surrogacy agent who helped coordinate the agreement between Ms Pattaramon and the Australian couple told her to abort the pregnancy after doctors learnt one of the twins had Down Syndrome.
Ms Pattaramon told the newspaper she refused the abortion because of her Buddhist faith.
Commercial surrogacy, in which a woman is paid to carry a child, is not permitted in Australia but couples are able to use an altruistic surrogate who receives no payment beyond medical and other reasonable expenses.
However, Surrogacy Australia said more couples choose to go overseas than find an altruistic surrogate at home, with 400 or 500 each year venturing to India, Thailand, the United States and other places to do so.