Hudson-Odoi, 19, was arrested on 17 May and released on bail.
The England international wrote on Twitter: "At a time where there are bigger things happening in the world right now, you may also be aware of serious allegations that were made against me.
I have stayed silent and assisted the police fully throughout their enquiries, as I knew the day would soon come when my name would be cleared.
Following a full and thorough investigation, the police have now confirmed they will take no further action.
In a statement, the Metropolitan Police said: A man arrested on Sunday, 17 May following an allegation of rape has been released with no further action.
The incident is therefore no longer being investigated by police.
Chelsea have declined to comment.
Hudson-Odoi, who has three England caps and in September signed a new five-year contract with Chelsea, was the first Premier League player to test positive for coronavirus.
He made a full recovery and returned to training with his Chelsea team-mates, who resume their season with an away trip to Aston Villa on Sunday, 21 June.
The forward added: I would like to use this platform to thank everyone who has stood by my side and supported me during this difficult period.
I have learned that being a footballer and playing for one of the best clubs in the world comes with a great responsibility, and going forward I will try and use my platform as a Chelsea player to be the best role model that I can be.