Rapp filed suit in NYC claiming he met the disgraced actor in 1986 while performing on Broadway in "Precious Sons." Rapp claims Spacey invited him, 14 at the time, to a party at his home on the East Side of Manhattan. Rapp alleges Spacey, without consent, made an unwanted sexual advance and grabbed his buttocks, lifted him onto a bed and laid on his body.

According to the docs, obtained by TMZ, Rapp claims he got scared and ran into the bathroom, and although Spacey tried to persuade him to stay ... he left. Rapp says he suffered psychological injuries, severe emotional distress, humiliation, fright, anger, depression and anxiety.

What's more ... Rapp claims the experience has forced him to step away from his work as an actor and feels he'll be compelled to spend tons of money for medical expenses to recover from the ordeal.

As we reported ... Rapp first made these accusations back in October 2017. Spacey responded by apologizing and coming out as gay.

BTW ... there's another plaintiff in Rapp's lawsuit but he's anonymous. He says he was also 14 and an acting student and alleges he had intercourse and oral sex with Spacey. In the suit, he claims he resisted Spacey's repeated attempts to sodomize him before he was able to flee Spacey's apartment.