WBC champion Fury and WBA, WBO and IBF world champion Usyk were due to step into the ring at Wembley on 30 April.
A deal could not be agreed, though, with both parties unable to come to an agreement over terms for a purse split in the rematch clause.
Usyk's promoter Alex Krassyuk pointed the finger at Fury for "putting so many obstacles in front of making the deal."
Mandatory challenger Daniel Dubois is instead seemingly next in line for Usyk, and Fury claims that is bout the Ukrainian wanted rather than facing him.
He said in a social media post on Thursday: "[You] tried all week to get out of it, begging for a rematch like a little girl. You got your rematch, then didn't even wanna fight at that.
"Always know that you was never man enough to tangle with the 'Gypsy King' ever in your life. You little 14-stone coward, and your full team know it as well, and all the lawyers have got the full information.
"Keep running. Fight [Daniel] Dubois at the Copper Box now, and always know that you or anybody else like you would never tangle with the Gypsy King!"