The British heavyweights had been due to clash in Manchester on April 5 but Whyte suffered a severe injury in a strength and conditioning session when his finger was crushed between two weights.
Whyte told Mail Sport: "I was in the gym working out, and everyone is meant to put their weights away. But someone didn't, and their weight rolled under the bench. I was doing my weights at the time, and when I went to put my weights down, I crushed my finger between the dumbbell and the ones the other person left out. It literally just crushed it."
The fighter explained that he had to have seven stitches to hold the wound together because the injury was so serious.
Whyte said: "I've had seven stitches just to help hold it in place because it's really deep. It's all the way down to the bone and to the nerve. They had to glue it closed and then add the stitches to sort of hold it in place. Also, the place it is in is difficult because it's really easy for it to reopen."
Whyte has been told by doctors that he cannot punch for up to four weeks and expressed his "annoyance" at missing out on the chance to box in the UK for the first time since 2022.
He said: "I was over the moon with the fight and ready to take on a good opponent. I would have knocked him out, and it would have been a good statement. I wouldn't just walk away from that. Obviously, these things happen, but it's really annoying."
A replacement opponent is now being sought for Joyce ahead of the clash next month.
Queensberry Promotions, which is staging the event, said in a statement: "Due to a hand injury sustained during training camp, Dillian Whyte will no longer be able to compete in his scheduled bout on April 5 against Joe Joyce at the Co-Op Live in Manchester.
"A replacement opponent is now being finalised for 'The Juggernaut' and an announcement regarding this will be made shortly."