Tillerson also took aim at Russia's suspected role in the Syrian conflict.
"Russia has really aligned itself with the Assad regime, the Iranians and Hezbollah," Tillerson said at a meeting of G-7 foreign ministers in Lucca, Italy, on Tuesday.
"We want to create a future for Syria that is stable and secure. And so Russia can be a part of that future and play an important role, or Russia can maintain its alliance with this group, which we believe is not going to serve Russia's interest longer term," he said.
His remarks came just hours before his planned arrival in Moscow on Tuesday and amid growing criticism of Russia's support for the Syrian government after it launched a chemical attack last week that killed dozens of civilians, including small children.
Tillerson said the chemical attack against civilians made the Russians look "not so good," after Russia helped broker a deal that required Syria to get rid of its chemical weapons stockpiles in 2013.
"Stockpiles and continued use demonstrate that Russia has failed in its responsibility to deliver on its 2013 commitment," he said. "It is unclear whether Russia failed to take this obligation seriously or Russia has been incompetent, but this distinction doesn't much matter to the dead."
"It is clear to all of us that the reign of the Assad family is coming to an end," Tillerson said. "But the question of how that ends and the transition itself could be very important, in our view, to the durability, the stability inside of a unified Syria.
"We see no further role for the Assad regime longer term, given that they have given up their legitimacy with these types of attacks."
As the Trump administration gave mixed signals over the weekend regarding its strategy in Syria, Tillerson doubled down that America's priority in Syria and Iraq remains defeating ISIS.
ABC News' Devin Villacis and Ely Brown contributed to this report.