In an interview with Buzzfeed, Rapp said Spacey picked him up, put him on his bed and "was trying to get with me sexually" in 1986. Spacey was 26 at the time.
In response to the allegations, Spacey tweeted that he was "beyond horrified to hear [Rapp's] story."
Spacey tweeted: "I honestly do not remember the encounter, it would have been over 30 years ago.
"But if I did behave then as he describes, I owe him the sincerest apology for what would have been deeply inappropriate drunken behavior, and I am sorry for the feelings he describes having carried with him all these years."
Spacey, who has been famously private about his personal life, said that he now lives as a gay man -- which marked the first time the actor has made a public statement about his sexual orientation.
"This story has encouraged me to address other things about my life," Spacey said in his tweet. He continued that he has had "relationships with both men and women," and added, "I have loved and had romantic encounters with men throughout my life, and I choose now to live as a gay man."
His response to Rapp's accusation which came with a statement about his sexuality, led to criticisms about why Spacey would address those two matters together.
"Nope to Kevin Spacey's statement. Nope," tweeted Dan Savage, the "Savage Love" columnist. "There's no amount of drunk or closeted that excuses or explains away assaulting a 14-year-old child."
Ashlee Marie Preston, writer and activist tweeted: "I'm not interested in Kevin Spacey being gay. I'm curious as to why we're conflating sexuality w/ the molestation of a minor. Stay on topic."
Allegation against Spacey
Rapp, an actor with a long Broadway career who is now a regular on the TV series "Star Trek: Discovery," was a child actor in 1986 when he says he met Spacey. They were both performing on Broadway shows, he told Buzzfeed. They met at a post-show gathering and Spacey invited him to a party at his Manhattan apartment that was happening a few days later.
Rapp attended the party alone, but got bored because everyone else there was an adult, and he did not know anyone. He told Buzzfeed he went into a bedroom and watched TV past midnight. Rapp said he realized that the other guests had left the party when Spacey appeared at the doorway and approached him.
Rapp said his initial impression when Spacey came in the room was that he was drunk.
"He picked me up like a groom picks up the bride over the threshold. But I don't, like, squirm away initially, because I'm like, 'What's going on?' And then he lays down on top of me."
"He was trying to seduce me," Rapp told the website. "I don't know if I would have used that language. But I was aware that he was trying to get with me sexually." He said he felt Spacey "pressing into me" and "tightening his arms."
Rapp said he was able to "squirm" away and left the apartment.
Rapp told Buzzfeed that he had disclosed details of the encounter in 2001 in an interview with the Advocate, but he did not name Spacey on the record in that account. The Advocate's executive editor at the time, Bruce Steele, told Buzzfeed that Rapp had been talking about Spacey in his account.
CNN has not independently verified the allegations and has reached out to both Rapp and Spacey's representatives.
Why Rapp spoke out
Rapp began acting from the age of 9. He was one of the original cast members of the smash hit musical "Rent."
Since his alleged encounter more than three decades ago, Rapp said he had hoped he would never see Spacey again.
Rapp, who is now 46, released this statement through his publicist.
"I came forward with my story, standing on the shoulders of the many courageous women and men who have been speaking out, to shine a light and hopefully make a difference, as they have done for me.
Everything I wanted to say about my experience is in that article, and I have no further comment about it at this time."
The allegation against Spacey comes in the wake of a New York Times story this month detailing numerous accusations of sexual misconduct against powerful movie producer Harvey Weinstein. More than 40 women have accused Weinstein of inappropriate behavior, ranging from sexual harassment to rape. Through a spokeswoman, Weinstein has "unequivocally denied" all allegations of rape.
Since then, other powerful men have been accused of sexual harassment and misconduct.