The 30-year-old Simon withdrew with a neck injury, having also retired from the Italian Open in Rome last week because of a similar problem.
The French Open starts on Sunday and world No 13 Simon admits he is concerned about his fitness.
He told the tournament’s official site: “It is progressing well but too slowly.
“I was always very bad, too bad to play. I’ll leave Nice tomorrow morning and I’ll go and have an MRI. I did the same place as usual so it worries me a little.”
Australia’s Kyrgios also pulled out with injury, just four days from the start of the French Open.
Kyrgios, who knocked Rafael Nadal out of Wimbledon last year and defeated Roger Federer on clay in Madrid this month, retired from his second round match against Dominic Thiem of Austria with an elbow injury.
“I expected a very tough match and I’m really sorry for Nick,” said Thiem, who was leading 4-3 at the time.??
“But I have to take it like this and of course I’m happy to be in the quarter-finals.”
Thiem goes on to face defending champion Ernests Gulbis.
Lucky loser Quentin Halys from France replaced Simon in the second-round draw and faced Australian James Duckworth for a quarter-final spot, but lost 7-6 (7-0) 7-6 (7-5).
Leonardo Mayer advanced to the last eight with a 6-4 6-0 win over French wild card Lucas Pouille to set up a meeting against eighth seed Juan Monaco.
Ernests Gulbis beat Alexandr Dolgopolov 7-5 4-6 6-1, while Borna Coric ousted qualifier Gianni Mina 7-6 (7-4) 6-3.
Dominic Thiem progressed when Nick Kyrgios retired with an elbow injury while trailing 4-3 in the first set and Dusan Lajovic overcame Pablo Carreno Busta 6-3 4-6 7-5 to set up a meeting with American John Isner in the next round.