Mr Rapp, now 50 and also an actor, filed the legal action in September 2020 and had been seeking around $40m (£36m) in damages.
Mr Spacey, an Oscar winner, has denied the accusations against him.
He is separately facing five charges in the UK of sexual assault, to which he has pleaded not guilty. That trial is due to begin in June 2023.
On Thursday, following a three-week civil trial in a Manhattan federal court, a jury found that Mr Rapp had not proved his claim that Mr Spacey had made an unwanted sexual advance.
Deliberations lasted for more than an hour and, after the verdict was read out by the judge, Mr Spacey reportedly hugged his lawyers before leaving the courtroom.
Mr Rapp says that Mr Spacey was around 26 or 27 years old when he met him at a party at his Manhattan apartment.
In a Buzzfeed article from October 2017, Mr Rapp said that Mr Spacey had picked him up, placed him on a bed and lay down partially on top of him. “I was aware that he was trying to get with me sexually,” he wrote.
Taking the stand in his own defence in the third week of his trial, Mr Spacey said he was shocked when Mr Rapp went public with his allegations five years ago.
“I didn’t know how this could possibly be true,” the 63-year-old said, adding he would not have been sexually interested in Mr Rapp as he was underage.
Mr Rapp’s lawyer Richard Steigman argued there were gaps in Mr Spacey’s memory and changes in his recollection.
Mr Spacey’s lawyer Jennifer Keller countered that Mr Rapp’s story was a fabrication, and she said he was “getting more attention in this trial than he has in his entire acting life”.
Multiple allegations of sexual misconduct derailed the Hollywood star’s stage career. Netflix dropped him from its political drama series House of Cards and Christopher Plummer replaced him in the role of J. Paul Getty in All the Money in the World.
In August, a judge ruled that Mr Spacey must pay $31m to the producers of House of Cards for the costs they incurred after firing him from the show.
Mr Spacey won Oscars for his performances in American Beauty and The Usual Suspects.
In July, he entered a not guilty plea in London to criminal charges of sexually assaulting three men over a decade ago.
During the New York trial he was asked about a suggestion by Mr Rapp that he was a “fraud” for not coming out as gay sooner. Mr Spacey said he had always been private about his personal life, including his childhood, because his father had held racist and homophobic views.
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