The Spaniard had looked in prime form in his first two matches, losing just nine games against Sam Groth and Facundo Bagnis, but he revealed at a hastily-arranged press conference that he would not be able to continue.
A dejected-looking Nadal said he first felt pain during his quarter-final win over Joao Sousa in Madrid and the problem had continued to get worse.
He said: “I arrived here with a little bit of pain but it was something I was able to manage, but every day it was a little bit worse. We spent a lot of hours here working so hard to try to play.
“Yesterday I played with an injection, with anaesthetic, and I could play but yesterday night I started to feel more and more pain and this morning I felt I couldn’t move the wrist.
“I had a scan and the results are not positive. It’s not broken but if I keep playing it’s going to be broken the next couple of days.”
Nadal said the problem was to do with a tendon sheath and if it had not been Roland Garros, where he had hoped to become the first man ever to win 10 singles titles at a grand slam, he would not have started the tournament.
He said: “I’m thinking about winning the tournament and for that I have to play five more matches. The doctor says that is 100 per cent impossible.
“That’s part of life, part of my career. I’m going to keep going hard to recover as quick as possible.”
The Spaniard has had a catalogue of serious injury problems during his career, with his knees causing the most trouble, but he is optimistic this will not be a long lay-off and that he will be fit for Wimbledon in a month’s time.
He said: “We’re going to work hard to be ready for Wimbledon. I need a couple of weeks (with the wrist) immobilised. Then we’re going to do the treatment and we hope that works well.
“We expect to recover quick and to be ready for Wimbledon.”
Fellow Spaniard Marcel Granollers, who Nadal had been due to face, receives a walkover into the fourth round, where he will play either Dominic Thiem or Alexander Zverev.
This is Nadal’s earliest departure from the French Open, with the 29-year-old’s only two defeats at Roland Garros coming in the fourth round in 2009 and the quarter-finals last year.