The law, which parliament passed in June, bans "propaganda of non-traditional sexual relations" and imposes fines on those holding gay pride rallies.
It has attracted international condemnation and cast a shadow over the athletics world championships in Moscow, with questions raised over whether it will apply to athletes and spectators at next year's Winter Olympics in the Russian Black Sea resort of Sochi.
The International Olympic Committee is seeking clarification from Russia while there have already been some calls for a boycott of the Games.
Mutko told reporters before the start of the track and field championships that critics should "calm down", saying the rights of all athletes competing in Sochi will be respected.
On Sunday, at a news conference before the start of the final day of the August 10-18 championships, he blamed continuing debate on "an invented problem" in Western media.
"We don't have a law to ban non-traditional sexual relations," he said. "The mass media in the West have focused much more on this law more than they do in Russia."
Critics of the anti-propaganda law have said it effectively disallows all gay rights rallies and could be used to prosecute anyone voicing support for homosexuals.
Mutko said the law was intended to protect Russian children.
"We want to protect our younger generation whose physicality has not been formulated. It is a law striving to protect rights of children - and not intended to deprive anybody of their private life," he added.
Few athletes at the world championships have openly talked about the legislation, although Russia's world pole vault champion Yelena Isibayeva caused international uproar when she spoke out in favour of it and appeared to condemn homosexuality, before later backtracking and saying she had been misunderstood.
American 800 metres silver medallist Nick Symmonds branded her as "behind the times", while Swedish high jumper Emma Green-Tregaro made a gesture of support for Russia's gay community during competition by painting her fingernails in the colours of the rainbow flag used by the gay movement.
After being warned the gesture broke the sport's code of conduct, Green-Tregaro appeared in Saturday's final with her rainbow nails changed to red.
Mutko, without referring to Green-Tregaro, said he hoped athletes in Sochi "come to compete and don't have time for other things".
He reiterated that athletes' private lives in Sochi would be safe.
"Russian athletes, foreign athletes, guests, those who come to Sochi will be granted all rights and freedom," he said. "This law does not deprive any citizen of rights, whether athletes or guests."
(Editing by Alison Williams)
