Russia has come under international criticism from human rights and gay groups after passing legislation outlawing the promotion of homosexuality.
Bach said he again raised the issue with Sochi organizers while attending the Olympic flame lighting ceremony at Ancient Olympia in southern Greece. The German Olympic fencing champion was elected to head the International Olympic Committee on Sept. 10.
"The task of the IOC is to ensure the application of the Olympic Charter at the Olympic Games and for all the participants of Olympic Games. That means we have to ensure a policy of non-discrimination at all levels, for all athletes and for all other participants," Bach said in brief remarks after the ceremony.
"We have the assurances of the highest authorities in Russia, and yesterday I spoke with the Russian delegation here in Olympia again and they reassured us that the Olympic Charter will fully apply for all the participants of the games."
Gay rights for both athletes and spectators have been a constant issue for organizers of the Russian games, and police security was tight at Sunday's ceremony but no protest groups were present at the small southern Greek town.
In Athens, gay Greek activist Zak Kostopoulos said protests against Russia were not aimed at the games.
"Of course this has nothing to do with the athletes and (protests) should not affect the sporting events in any way," Kostopoulos said, speaking by telephone.
"But we consider this law to be totally unacceptable, along with the actions of (Russian President Vladimir) Putin. This is an effort to criminalize even a discussion about being gay."