Washington has stepped up the pressure on Russia to rein in Assad after a suspected chemical weapons attack on a rebel-held Syrian town that killed at least 87 civilians and triggered retaliatory US missile strikes on a Syrian air base.
As G7 foreign ministers urged a diplomatic push to end the six-year conflict and create a lasting peace for a unified Syria, Tillerson said: "Our hope is Bashar al-Assad will not be part of that future."
"To be clear, our military action was a direct response to the Assad regime's barbarism," Tillerson said at the G7 meeting in Italy.
"The United States' priority in Syria and Iraq remains the defeat of ISIS," he added, referring to the so-called Islamic State (IS) group.
Moscow said it was hoping to avoid confrontation and engage in "constructive cooperation" with Washington, as Tillerson headed to the Russian capital on the first visit by a senior member of President Donald Trump's administration.
"At the moment there is no consensus on new sanctions as an effective instrument," Italy's Foreign Minister Angelino Alfano said, after his British counterpart Boris Johnson had raised the issue.
But after talks in the Tuscan city German Foreign Minister Sigmar Gabriel said: "All the G7 states want to avoid a military escalation and want a political solution without a new spiral of violence.
"We want to bring Russia around to supporting the political process for a peaceful resolution of the Syrian conflict."
The US has warned Damascus that further use of chemical weapons could bring fresh retaliation.
US warships in the Mediterranean on Friday fired 59 Tomahawk missiles at the Shayrat air base near Homs, destroying an airfield from which Washington believes Assad's jets launched the deadly chemical attack.
The strike was the first time Washington had intervened directly against the regime of Assad, who is fighting a six- year civil war with the backing of Russia and Iran.
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