The Scot had been due to take on the home favourite for a place in the last four but play at Roland Garros was disrupted by downpours in Paris.
Murray and Gasquet will now try again on Wednesday.
Defending champion Stan Wawrinka, who was scheduled to play Albert Ramos-Vinolas in the quarter-finals, will also have to wait until Wednesday to play.
In the women’s draw second-seeded Agnieszka Radwanska is out after losing to Tsvetana Pironkova 2-6, 6-3, 6-3 in the fourth round.
Radwanska, who held a 6-2, 3-0 lead when her match resumed on Court Suzanne Lenglen, lost 10 consecutive games to put herself in deep trouble.
She asked for a medical timeout after the third game of the decider to receive treatment on her right forearm and hand. It didn’t have an immediate effect as she dropped her serve at love in the next game.
Radwanska finally managed to stop the downhill spiral when she broke in the fifth game but could not stop Pironkova from advancing to the quarter-finals.