Robert, 53, scaled the 210-metre Tour de Montparnasse without a safety harness in just below an hour on Tuesday.
During the climb he tried to fix the Nepalese flags and prayer flags to the building but they failed to stay attached.
Before setting off he said that his thoughts were with mountaineers stranded at the Everest base camp, where many people died in avalanches set off by the quake.
Robert has climbed more than 130 structures all over the world, never using ropes or safety equipment.
French Foreign Affairs Minister Laurent Fabius said that 560 French nationals, of the 2,050 who were in Nepal at the time of the quake, have still not been located.
Two French tourists were reported dead on Monday and Fabius said that there is reason to believe that another was killed in an avalanche and his partner injured.
About 20 French citizens have been injured.
The latest death toll on Tuesday afternoon stood at 5,057 with more than 10,000 injured.
Prime Minister Sushil Koirala told the Reuters news agency that the final toll could be as high as 10,000.