Rapp told how Spacey laid down on top of him in a bedroom during a party at the older man’s flat in 1986.
The Double Oscar-winner said he was “beyond horrified” to hear the story and did not remember the encounter.
‘Hand on thigh’
Speaking to the programme, John said he first met the Broadway actor in 1984 at a summer theatre when he was 16.
The pair exchanged letters and, a year later, Spacey invited him to spend a weekend at his home in New York City.
John told how a “charming and brotherly” Spacey showed him around the city, took him out for dinner and introduced him to famous friends on their weekend together.
On the first evening, Spacey talked a lot about his work and became affectionate “in a way I certainly wasn’t interested in”, he said.
He put his hand on John’s thigh, an arm around his shoulder and rubbed his arm.
At bedtime, John said Spacey asked him to share his bed, but John insisted on sleeping on the sofa.
“It was an icy ‘goodnight’ with the lights off. I thought I was going to be kicked out in the morning,” he said.
“As we went to sleep, he was sobbing from his bed,” said John.
He said he felt it was attention-seeking behaviour in an attempt to get him to respond.
The following morning, John said he woke up with Spacey’s head on his stomach and his arms wrapped around him.
“He was in his underwear, I was fully clothed. I supposed it was some sort of New York theatre actor ‘good morning,'” he said.
‘Stupid or predatory’
After a second day enjoying New York, John said once back at the flat, Spacey became affectionate again and told him he felt “misunderstood”.
“I burst into tears because I couldn’t articulate any more what was happening to me. I was scared,” said John.
“To his credit, he backed off and we went to sleep.”
John said he is telling his story now after reading about Rapp’s similar experience and he wants young people to know they should be vigilant and speak out.
“It seems he was grooming me,” John said.
“For me, I never let on that that’s what I was interested in. I never discussed it, nor did I want it.
“I was uncomfortable at best, traumatised at worst.
“He was either very stupid or predatory – or maybe a little of both.”
John points out that neither of them drunk any alcohol that weekend.
He said he didn’t tell his parents or the authorities at the time because he thought Spacey’s behaviour might have been “permissible” and worried he might have “given off a vibe” that he was interested in Spacey.
The BBC is seeking a response from Spacey.
In light of Rapp’s allegations, Netflix has suspended production of political drama, House of Cards, in which Spacey stars.
Meanwhile, the Old Vic, a London theatre where Spacey worked for 11 years, has set up a confidential complaints process for people involved with the theatre.