Gasquet appeared to hurt his lower back after stretching for a shot at 3-2 down.
He called for the trainer to come to Court One but the contest was over before assistance arrived.
Seventh seed Gasquet was wincing in pain but carried on.
After hitting a return into the net in the next game, Gasquet knew he could not continue and went to shake hands with number 12 seed Tsonga, at 4-2 and 40-0 down.
Gasquet apologised to the crowd as he went off and Tsonga helped him carry his bags from the court.
Tsonga will play either British second seed Andy Murray or 15th seed Nick Kyrgios of Australia in Wednesday's quarter-finals.