The 58-year-old apologised "for what would have been deeply inappropriate drunken behaviour" and said that he had used the allegation "to address other things about my life".
Season six is due to arrive on Netflix in the middle of next year, and the streamer said it had always intended for it to be the show's last run even prior to this week's allegations against its star.
The Netflix drama, which stars Spacey and Robin Wright as ambitious Washington political couple Frank and Claire Underwood, will end in 2018 with 13 episodes.
Rapp, now 46, claims Spacey then climbed on top of him and made sexual advances before he managed to "wriggle free".More news: Rainy and windy, high near 50 — TODAY'S FORECAST
More news: HSBC profits climb as Asia shift pays off
More news: Kim Jong-Un's wife makes rare public appearance
Spacey has since issued an apology for the encounter, which he says he does not remember, although he has been heavily criticised for combining the apology with an announcement that he's now living "as a gay man". He also spoke publicly for the first time about being gay, which draw backlash from some observers as an attempt at deflection.
Spacey has come under fire around the industry for the alarming allegations against him as well as the way in which he responded to them.
Spacey's work on the series has also earned him five Emmy nominations for outstanding actor in a drama.
"I created a House of Cards 30 years ago and up to this point it has been an exhilarating and totally joyous journey", Dobbs told Deadline.
He went on to state: 'That said, I take reports of such behavior seriously and this is no exception. The avalanche of allegations began earlier this month after the New York Times published a story alleging that producer Harvey Weinstein had sexually harassed numerous women. He had never disclosed his sexuality before but said Rapp's story encouraged him to speak.