Kim used a cast along with ice therapy and other treatment to make it through the Open Championship at Hoylake last week after injuring his ankle when he slipped at his rental house after the first round.
After deciding not to withdraw, Kim shot three straight rounds in the 60s to tie for second with Jason Day, Jon Rahm and Sepp Straka behind runaway winner Brian Harman.
"I am very gutted to not be able to defend my first PGA Tour win," Kim said in a statement released by Wyndham organisers on Wednesday.
"I am still having a hard time with my ankle so I am getting the treatment I need to heal completely."
Kim produced a stunning nine-under-par 61 in the final round to earn his first PGA Tour title at Sedgefield Country Club last year.
He went on to win a second title at the Shriners Children's Open, becoming the first player since Tiger Woods to win twice on tour before turning 21.
He also starred for the International Team at last year's Presidents Cup.
