Rune, who is 16th in the world rankings, was set to make his Olympics debut in Paris.
"I am really sorry that I will be unable to play the Olympics. It's something I've been looking forward to playing and being a part of," Rune wrote on X.
"I have played with pain in my wrist throughout both the clay and grass seasons, so I have to take the medical recommendations seriously."
Rune most recently competed last week in Hamburg, Germany where he retired from his quarter-final match against eventual champion Arthur Fils of France while trailing 6-4 4-1.