The law "creates far more problems than it will ever solve, and it leaves the underlying problem untouched," said Ernie Chambers, the only lawmaker in the Nebraska state legislature to vote against the bill.
"I do not believe that a woman -- young, middle-aged or old -- just on a whim says 'I think I'll give up my baby'," Chambers said.
The law, which took effect on July 18, says: "No person shall be prosecuted for any crime based solely upon the act of leaving a child in the custody of an employee on duty at a hospital licensed by the state of Nebraska."
Sarah Ann Lewis, policy coordinator at the Voices for Children in Nebraska rights advocacy, expressed concern for parents' rights.
"The law says 'an adult' can abandon the child. It doesn't even have to be a parent," she told AFP.
"Hopefully the system would kick in and parents be notified that their child has been abandoned and the family could be reunited, but one of our specific concerns involves the possible deprivation of the parents' or one parent's rights," she said.
In the state of Nebraska, a dependent child is defined as anyone under the age of 19 who is living with a relative or with a caretaker who is the child's legal guardian or conservator, Lewis said.
Chambers slammed the law for treating the symptoms and not the cause of child abandonment.
"What this society needs to do is consider the causes that might compel a woman to give up what might mean more to her than anything in her life," he said.
"We need sex education. We need to stop talking this abstinence nonsense based on religion and look at the reality of what these women face."
Nebraska is the last of the 50 US states to pass such a law, widely known "safe haven" legislation.
It is also the only state to extend the reach of the law to include children up to the age of 19, according to Chambers and Lewis.
Lawmakers who voted in favor of Nebraska's safe haven bill were not immediately available for comment.