The suit also targets the organization's secret files of Scouts leader abuse allegations, and a judge has ordered the Scouts to turn over documents covering 20 years.
In addition, the civil suit says the Scouts should have known a troop leader put a boy in jeopardy in the 2007 criminal case, the Los Angeles Times reported.
The newspaper is withholding the name of the family to protect the boy's identity.
When first told of the allegations, the boy's mother said, a Scouts executive hesitated to inform law enforcement.
"He said that wasn't necessary, because the Scouts do their own internal investigation," the woman told the Times. "I thought that was really weird. ... I thought it was really important to call the sheriff right away."
The mother called in police herself. Scout leader Al Steven Stein, then 29, was charged with abusing her son and two other boys. He pleaded no contest in 2008 to felony child endangerment.
Stein was put on probation, but when officials later found pictures of nude children on his cellphone data card, he was sent to prison.
In addition to damages, the civil suit successfully requested a court order forcing the Scouts to hand over thousands of confidential files on allegations of sex abuse by Scout leaders and others across the country, saying they show a "culture of hidden sexual abuse."