Comments by Vitaly Mutko run against earlier talk that the Games could be exempted from the new rule, which provoked heavy criticism from the West and triggered a growing boycott of Russian vodka in the United States.
"No one is forbidding an athlete with non-traditional sexual orientation from coming to Sochi, but if he goes onto the street and starts propagandising it, then of course he will be held accountable," Mutko was quoted as saying by RIA news agency.
Foreigners found to have broken the law can be deported from Russia, in addition to being fined up to 100,000 roubles (£1,983) or held for up to 15 days.
The law, as well as a ban on adoptions of children by same-sex couples, are part of a more conservative course taken by President Vladimir Putin on social issues since his return to the Kremlin in May, 2012.
Putin critics say the law is one in a string of repressive measures introduced by the former KGB spy during more than a year of his third presidential term to crack down on dissent.
Putin, who denies clamping down on political opponents, has made Sochi a top priority for Russia to help its image abroad by propagating it as a modern state with top-notch infrastructure.
But the latest controversy only adds to criticism over cost overruns and accusations of widespread corruption marring the February 7-23 games.
(Reporting by Gabriela Baczynska; Editing by Michael Roddy)
