"Russia's failure to resolve the chemical weapons issue in Syria calls into question its relevance to the resolution to the overall crisis," said Tillerson, who called on Russia to stop using its veto power to block United Nations Security Council actions related to the Assad regime's use of such weapons.
Tillerson was addressing a meeting of high-ranking diplomats in Paris as part of a new international effort to address the use of chemical weapons.
Most recently, Tillerson said more than 20 civilians, most of them children "were victims of an apparent chlorine gas attack" on Monday in the Eastern Ghouta suburb of Damascus, Syria. "Whoever conducted the attacks, Russia ultimately bears responsibility for the victims in East Ghouta, and countless other Syrians targeted with chemical weapons since Russia became involved in Syria."
"The recent attacks in East Ghouta raise serious concerns that Bashar al-Assad's Syrian regime may be continuing its use of chemical weapons against its own people," he said.
To those who deploy chemical weapons, Tillerson had this message: "You will face a day of reckoning for your crimes against humanity, and your victims will see justice done."
As Tillerson was speaking in Paris, US Ambassador to the United Nations Nikki Haley put out a statement directly tying the Eastern Ghouta attacks to the Assad regime, calling them, "yet another demonstration of (the regime's) blatant disregard for international law and cruel indifference for the lives of its own people."
Haley is expected to speak at an open meeting of the UN Security Council on chemical weapons in Syria on Tuesday afternoon. Two diplomats tell CNN Russia called for the session.
Last week, Tillerson gave a speech at Stanford University, in which he outlined the central tenets of the Trump administration's policy in Syria. In those remarks, he reiterated US support for a UN-led political process the Trump administration believes will ultimately result in Assad's removal from power.
Nevertheless, Tillerson and others have made clear that defeating ISIS is the administration's primary goal in Syria, and the continued US troop presence in the country was tied to that effort, and not to Assad's removal.
Still, the US is hoping it can persuade the Russian government -- Syria's most important ally -- to bring the long-beleaguered but still-determined Assad to the negotiating table.
Assad has waged a nearly seven-year-long civil war against Syrian opposition groups, which has decimated large parts of the country and created a humanitarian catastrophe of historic proportions.
While the US has long criticized the Assad regime for its brutal tactics, including the alleged use of chemical weapons against civilians, it has only taken limited action against his government. Most notably, the Trump administration launched Tomahawk cruise missiles at a Syrian airbase last April over reports of a chemical weapons attack in northwestern Syria.
In 2013, then-President Barack Obama said the regime had crossed a "red line" with its use of such weapons, including sarin gas. But his administration ultimately opted not to intervene militarily after Russia stepped in to arbitrate an international agreement aimed at removing all chemical weapons from the country.
But in the intervening years, the Syrian government has been accused of additional chemical weapons attacks, including with chlorine gas.
"There is simply no denying that Russia, by shielding its Syrian ally, has breached its commitments its agreements to the United States as a framework guarantor," Tillerson said Tuesday. "It has betrayed the chemical weapons convention and UN Security Council resolution (2118)."