The Austrian star sustained the injury during a second-round match against Adrian Mannarino at the ATP event in Mallorca earlier this week.
Thiem, who had already opted out of competing at the Tokyo Olympics, will have to wear a wrist splint for five weeks.
He is now in a race against time to be fit for the defence of his first grand slam title at the US Open at the end of August.
The 27-year-old said on Thursday: “I’m going to do everything the doctors say in order to recover as quickly as possible.
“They’ve informed me that I might be out for several weeks but I will do my best to be back on court soon.
“I’m really sorry for pulling out of the upcoming three tournaments I had in my calendar: Wimbledon, Hamburg and Gstaad.
“They are very important tournaments for me. I appreciate all the support from the fans in these difficult moments – I’m determined to come back stronger.”
Fans were left shattered over the sad news.