Given this, the US attack on Syria last week in retaliation for what it said was a chemical weapons attack by Assad's forces on Syrian civilians confounded many diplomats.
Tillerson said Russia's support of the Syrian regime made it complicit in the Assad's actions.
Syrian President Bashar al-Assad can not remain as leader, and he must be held to account for perpetrating war crimes against his own people, Prime Minister Justin Trudeau said today.
Though such comments hint at a more activist US foreign policy focused on preventing humanitarian atrocities, Trump has consistently suggested he prefers the opposite approach.
He said the United States was rededicating itself to hold to account "any and all" who commit crimes against innocent people.
The G7 gathering in the Tuscan city of Lucca brought foreign ministers from the United States as well as Britain, Canada, France, Germany, Italy and Japan to the 15th century Ducal Palace.
"I hope Russian Federation is thinking carefully about its continued alliance with Bashar al-Assad, because every time one of these horrific attacks occurs, it draws Russian Federation closer into some level of responsibility", Tillerson said on ABC's "This Week with George Stephanopoulos" on Sunday.
The ministers will also talk about growing tensions with North Korea, with the United States moving a navy strike group near the Korean peninsula amid concerns over Pyongyang's nuclear ambitions.
World leaders have long disagreed on how to handle the Syrian civil war, a six-year catastrophe that has pitted Assad's regime, multiple anti-Assad rebel groups and the Islamic State (ISIS) militant group against each other.
Successive attempts by top Trump administration officials to articulate a plan have only furthered the appearance of a policy still evolving, even after the USA broke with precedent last week by attacking Assad's forces.
Syria, though, topped the agenda.
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Japan's Foreign Minister Fumio Kishida said he told Tillerson that Tokyo supports the U.S. in its push to "deter the spread and use of chemical weapons", and discussed the pressing North Korean nuclear threat.
However, the foreign ministers will talk about Libya.
Russian Federation planned to put forward a proposal on Monday for an independent and impartial investigation of the attack, a spokesman for German Foreign Minister Sigmar Gabriel said, calling it "a good and important sign".
Moscow should end its "toxic" support for Assad, Johnson said in a statement quoted by AFP.
Russia was kicked out of the club of industrialized nations, formerly the G-8, after its 2014 annexation of Ukraine's Crimea region and assistance for pro-Russian separatists in eastern Ukraine.
President Donald Trump's surprise missile strike against Syria may have unsettling geopolitical ramifications, but politically speaking, it appears to be a thunderous success for the president among the American people.
But Tillerson was more equivocal, saying that the priority was the defeat of ISIS.
The mixed messages have confused and frustrated European allies, who are eager for full US support for a political solution based on a transfer of power in Damascus.
USA ambassador to the UN Nikki Haley suggested regime change in Syria was inevitable in the wake of last week's chemical weapons attack that was widely blamed on the Assad regime. And it has has ravaged the nation: About 400,000 Syrians have died in the fighting, according to humanitarian groups, and almost 5 million people have fled the nation as refugees, contributing to an worldwide refugee crisis with many European nations struggling to cope with an influx of migrants. "If you look at his actions, if you look at the situation, it's going to be hard to see a government that's peaceful and stable with Assad", she said.
The session on the sidelines of the Group of 7 summit in Italy was to include Middle East countries, including Turkey, Saudi Arabia, Jordan and the United Arab Emirates, that share a USA interest in resolving the conflict and resisting Iran's influence in Syria.
Italian media said the aim was "to avert a unsafe military escalation".