Rubio blamed Russian President Vladimir Putin for enabling the attacks.
"Our expectation is that the Russian Federation does its part to protect its own citizens in full respect of human rights principles," the European Union's foreign policy chief said Monday at a Russian news conference.
Foreign Minister Sergey Lavrov is the latest of several officials claiming the widespread reports aren't true.
While at an event at New York City's LGBT center, Clinton slammed Trump's silence on Chechnya.
Putin and Chechen authorities say gay men have not been tortured and killed, but journalists in Russia are sticking to their reports.
"If my family knew that I was gay, they would have killed me long ago," an anonymous gay man from the Russian province explained to AFP.
Nikki Haley finally addressed the gay murders and torture in Chechnya, and while the U.N. ambassador's statement is more than we got from Trump or Tillerson, it feels too little, too late.