Canivet was appointed earlier this year to lead the IOC probe after revelations of widespread doping in Russia and systematic cheating during the 2014 Games by Russian athletes, who were allegedly helped by the country's secret service.
IOC spokesman Mark Adams said Canivet, a French Constitutional Court judge, had stepped down for "strong personal reasons".
Canivet offered to remain at the IOC's service and "give advice whenever needed", Adams said.
His investigation was focusing on allegations of government involvement in Russian doping during the Sochi Games.
The 69-year-old Schmid is a former president of the Swiss Confederation and ex-head of the federal council in charge of the army, population and sport.
Canivet's resignation comes just days before Canadian lawyer Richard McLaren publishes the second part of his report into Russian doping in London on Friday.
His first report in July triggered a partial ban of Russian athletes at the Rio de Janeiro Games.
(Reporting by Karolos Grohmann; editing by Ken Ferris)
