"We want to see Russia cease selling arms to the Assad regime," Clinton said in an interview with CBS News Thursday.
Earlier this month, Russia joined the growing international condemnation of the violence in Syria after government forces stepped up their crackdown on protesters as the holy Islamic month of Ramadan began.
Russian President Dmitry Medvedev said Russia may make "some kind of decisions" against the Syrian government if Assad does not restore peace and initiate reforms, RIA Novosti reported.
Asked by CBS News whether the United States would urge Assad to resign, Clinton said, "I think we've been very clear in what we have said about his loss of legitimacy. ... We are, I think, building the chorus of international condemnation. And, rather than [the United States] saying it and nobody else following, we think it's important to lead and have others follow as well."
Clinton said countries needed to pressure Assad through sanction on the country's oil and gas industry.
"[We] want to see Europe take more steps in that direction. And we want to see China take steps with us. We want to see India, because India and China have large energy investments inside of Syria," she said.