Nationally circulated newspapers warned that the anonymous drop-off, known as "Stork's Cradle," is open for abuse and could traumatize youngsters. They also condemned the father, saying his action could spur copycats.
The drop-off was opened last Thursday by the Catholic-run Jikei Hospital in the southern city of Kumamoto to discourage abortions and the abandonment of children in unsafe public places. The same day, a boy believed to be 3 was found inside.
The boy, who was in good health, reportedly said he was dropped off by his father, who was seen holding the youngster's hand as they approached the hospital. They apparently rode Japan's bullet train to Kumamoto, but it was unclear where they lived.
"I came with Daddy," the boy was quoted as saying by the Mainichi newspaper.
The find triggered a wave of outrage among political leaders, with Prime Minister Shinzo Abe saying "Anonymously throwing out a child is unacceptable." He urged parents to consult social workers for help if raising children gets too tough.
Local media reported that the boy was able to identify himself by name. But it was unclear whether the father had been identified.
The hospital has refused to comment on the case, citing privacy concerns, but said there were age limits on its drop-off service.
Police have decided no crime was committed in the current case because the child was left in a situation in which it was not exposed to immediate harm, Kyodo News reported Wednesday.
"We must rethink the meaning of the baby drop-off," the conservative Sankei newspaper said in an editorial. "Unlike a baby, a toddler may suffer from trauma."
"This little boy must be experiencing great loneliness. We urge his mother or father to come forward," the newspaper said, calling his abandonment "unforgivable."
The Yomiuri newspaper said it was too early to judge the baby-drop, but said that it must be used for its original purpose of receiving newborns, not young children. Parents should also be encouraged to seek outside help before dumping their offspring.
The Mainichi said the misuse of the box could inspire copycats.
A small hatch on the side of the hospital has been set up to allow people to drop off babies into an incubator 24 hours a day.
The drop box was created after a series of high-profile cases in which newborn babies were left behind in parks and supermarkets, triggering a public outcry and government warnings against abandoning babies.