Hay has informed the Professional Rodeo Cowboys Association that he has not healed quickly enough to compete in the National Finals.
"I needed to get more mobility in my leg and feel comfortable that I could be competitive," Hay said in a press release.
"If I'm going to go there, I want to be a threat and believe I have a chance. People pay a lot of money to go to Las Vegas, and they want to see that guys are competing to win. I didn't feel I would be at that level."
This will be the first NFR appearance for the 21-year-old Wright, the 2009 Rookie of the Year and younger brother of 2008 world champion Cody Wright.
Jesse Wright had finished just $428 behind No. 15 Sam Spreadborough on the money list and originally missed qualifying. But Hay's injury will give him a chance to compete in the NFR, which runs through Dec. 11 at the Thomas & Mack Center.
"I bet it was very hard for Rod [to withdraw]," said Wright. "I have never been in that type of situation, but I bet he thought about it a long time."