A clip of the video,
which features a gay couple holding hands, got the 18+ rating from the independent TV Rain channel
in Russia on Monday.
The channel told CNN that it didn't want to break the controversial law, which bans "propaganda of nontraditional sexual relations around minors" and bars public discussion of gay rights and relationships within earshot of children.
"There are no legal precedents for this law, so we just don't know what comes under this law and (what) doesn't," a TV Rain spokesperson told CNN.
"Therefore, fearing to break the law -- especially given the high attention to TV Rain from the supervising authorities -- we decided to put a marker (on the video)."
Clinton's video was released over the weekend to announce the start of her 2016 U.S. presidential campaign.
It features about five seconds of two men holding hands. One of the men says, "I'm getting married this summer to someone I really care about."
The former senator and first lady first declared her support for same-sex marriage
in early 2013, saying that "gay rights are human rights, and human rights are gay rights."
Russia's controversial law caused an international outcry
after it was passed by the Russian Parliament and signed by President Vladimir Putin in July 2013.
Human Rights Watch
described the anti-gay propaganda law as "a profoundly discriminatory and dangerous bill that is bound to worsen homophobia in Russia."
Rights campaigners called for a boycott of the 2014 Winter Olympics in Sochi, and a number of bars around the world stopped serving Russian vodka in protest.
U.S. President Barack Obama -- Clinton's former boss -- said at the time that he found the legislation offensive.
"I have no patience for countries that try to treat gays or lesbians or transgendered persons in ways that intimidate them or harmful to them," Obama told Jay Leno in 2013.
Putin defended the law, noting that unlike other countries, Russia decriminalized homosexual relationships (in 1993).
"We don't outlaw anything and don't nab anyone," he said before the 2014 Games. "That's why you can feel safe and free here," he added, "but please leave our children in peace."
The rights group ILGA-Europe said in a May 2014 report
that Russia was the worst place in Europe (out of 49 countries) for LGBTI people to live.