While he did not rule out the future use of military force, Tillerson said the USA administration was mindful of "the lessons of what went wrong in Libya when you choose that pathway of regime change".
Graham spoke several days after Trump unilaterally ordered a missile attack on a Syrian airbase in response to Assad's deadly chemical weapons attack on civilians and as the Trump administration tried to explain its foreign policy.
He did not offer any proof for that assertion.
The cruise missile strike came after a nerve gas attack on civilians that killed more than 80 people, including children, in the rebel-held Syrian province of Idlib at the start of last week.
But the secretary of state said when he meets with the Russian foreign minister this week, he will bring up Russia's obligation under a 2013 agreement to ensure the Syrian government got rid of its chemical weapons. "And we're not going to allow things like this to happen to innocent people".
The Russian Defense Ministry said on Wednesday that the airstrike near Khan Shaykhun by the Syrian air force hit a terrorist warehouse that stored chemical weapons slated for delivery to Iraq, and called on the UN Security Council to launch a proper investigation into the incident.
But as lawmakers called on Trump to consult with Congress, Trump administration officials sent mixed signals on the scope of future USA involvement.
On NBC News' "Meet the Press", Haley said that the strike sent a message to Russian Federation and other countries that support Assad's regime, including Iran.
"Here's what I think Assad is telling Trump by flying from this base: F-U", Graham said.
In his first televised interview, H.R. McMaster pointed to dual USA goals of defeating the Islamic State group and removing Assad from power. "And so we would want to appeal rationally to Russia", McCaster said on "Fox News Sunday".
"I believe that the United States of America can address both at the same time", McCain said.
Britain has announced that it "fully supported" the USA attack, and British Foreign Minister Boris Johnson canceled a visit to Moscow citing fundamental developments in Syria.
Syrian army forces had been losing ground across the country until Russian Federation intervened militarily in September 2015, propping up Assad and protecting its own interests in the region.
Tillerson said defeating the Islamic State group remains the top focus.
Tillerson stressed, however, that he has no "hard evidence" leading him to believe Russian Federation played an active role in the chemical attack.
Fallon wrote that the Kremlin was responsible "by proxy" as the "principal backer" of Assad's government.