The Gunners pulled out of the deal after failing to secure the services of another forward before the transfer window closed on Monday night.
Bendtner had travelled to the Palace training ground to discuss personal terms and undergo a medical, but he will now return to north London and he has vowed to do all he can to help Arsenal achieve success this season.
In a statement issued through his management company, the Dane said: "It's no secret that Arsenal and myself had agreed that I could leave the club during the 2013 summer transfer window subject to an agreement between all parties.
"An agreement had been reached with more than one club this week but as Arsenal were unable to secure the services of another striker, they reserved their right to not sanction any deal to be concluded.
"Naturally, I was very disappointed as I was looking forward to a new challenge, with a new club and a fresh start in English football.
"Many things have been said about where my future lies after many years at Arsenal; the truth is I just want to play football which is when I'm happiest and Arsenal is a fantastic environment for any professional to ply their trade.
"I still feel very strongly about Arsenal Football Club and, following positive talks with the manager, I am looking forward to working hard to regain full fitness and doing my very best to help to the team and the manager fulfil our objectives and ambitions this season.
"I've learnt a lot from my loan experiences at Sunderland and Juventus, and I want to assure all Arsenal fans that I will give everything I have to contribute to what I believe will be a successful season for the club."