“Cesar’s just, obviously, very happy, very emotional,” Mr. Mates said.

The jurors, who spent more than a week deliberating the 16 charges, said they struggled to stay impartial as they fought through the strong emotions stirred by the case. They used a white board to painstakingly account for the many details they were given and built a timeline. They debated the definition of “reasonable doubt” and had the judge redefine it for them.

Jury members fainted, experienced dizziness and stomach problems, and lost sleep, said Louise Corcoran, a juror who is a teacher from West Islip.

But in the end, they decided that Mr. Gonzales-Mugaburu was not guilty of any of the charges.

“We were, you know, waiting for the rest of the story,” Ms. Corcoran said. “It never came.”

Jurors had “to try to finish the story in your head, which you really can’t do,” she said.

Tim Carney, the jury foreman, who is a firefighter from Islip, also pointed to gaps in the case against Mr. Gonzales-Mugaburu. The prosecutors “just never built that bridge, in our eyes,” Mr. Carney said.

“We can’t ask for evidence to come in,” he said.

The case began in January 2016, when two children in Mr. Gonzales-Mugaburu’s care told a caseworker that he had made inappropriate comments to them. After he was arrested, others came forward saying he had sexually abused them. The case eventually involved eight boys, who are now 16 to 29; all of them testified.

The indictment against Mr. Gonzales-Mugaburu included charges of predatory sexual assault against a child, criminal sexual acts in the second and third degree, sexual conduct against a child and endangering the welfare of a child.

Some of the boys testified that they had reported abuse to social workers and police officers over the years, but earlier investigations had been quickly closed by the authorities. One witness, who is now 29, testified that when he lived at Mr. Gonzales-Mugaburu’s house in Ridge, the boys were told to lie in advance of visits from social service agency workers.

The 29-year-old returned to the courthouse on Tuesday. He said he was shocked that Mr. Gonzales-Mugaburu had been set free.

“Our fight, our testimony, just went down the drain,” he said, wiping tears from his eyes. “We’re not lying.”

Dari Schwartz, the bureau chief of the district attorney’s child abuse and domestic violence unit, said the office was surprised by the outcome. In a statement, Thomas J. Spota, the district attorney, said, “We are extremely disappointed with the Mugaburu verdict.”

Richard D. Emery, a lawyer representing four of the former foster children involved, said his clients would still pursue civil cases against SCO Family of Services, a nonprofit that had placed children in Mr. Gonzales-Mugaburu’s care under contracts with the New York City Administration for Children’s Services and the State of Washington.

“They feel insulted and undermined,” he said. “It’s too bad they can’t get justice in the criminal court, but they will get justice in the civil court.”

Rose Anello, a spokeswoman for SCO Family of Services, declined to comment.

In Ridge, neighbors were upset that Mr. Gonzales-Mugaburu had returned to his boarded-up home on Tuesday.

Mike Lange, a former Suffolk County police officer whose home is across from Mr. Gonzales-Mugaburu’s and who testified during the trial, said he used to see one of the boys left outside for hours at a time, even in winter. He said he had told the boys that if they needed food, they could come to him.

But his wife, Ann Lange, said the children were afraid of Mr. Gonzales-Mugaburu.

“Stunned. Stunned. Stunned by the outcome,” Ms. Lange said.

Arielle Dollinger reported from Riverhead, and Eli Rosenberg from New York. Marc Santora and Nikita Stewart contributed reporting from New York, and Susan C. Beachy contributed research.