The 12th-ranked Tsonga, who has not played since his semi-final loss to Stan Wawrinka at the French Open, picked up the injury in Paris.
He already withdrew from grass-court tournaments in Germany and the Netherlands in previous weeks.
Nottingham Open director Paul Hutchins said: “Tsonga has had to pull out of three tournaments now in the lead up to Wimbledon and I hope he recovers well so he can take his place at the Championships.”
The Nottingham Open starts on Sunday. Play begins at Wimbledon on June 29.