He said the United Nations should first investigate the attack.
Tillerson is expected in Moscow this week for talks with Russian officials.
Before heading to Russia, Tillerson told reporters it was unclear whether Moscow had failed to take seriously its obligation to rid Syria of chemical weapons, or whether it had merely been incompetent. But he says that distinction "doesn't much matter to the dead".
Tillerson was also hoping to unearth new information about Russia's role in the presidential election. "But I've never been more encouraged by the Trump administration than I am today".
Mr Putin said on Tuesday Russia would be urgently asking the global chemical weapons watchdog - the Organisation for the Prohibition of Chemical Weapons - to investigate the incident.
It was Tillerson who, on March 30, signaled that Assad's removal was no longer a US priority when he said that the "longer-term status of President Assad will be decided by the Syrian people".
He then continued: 'Clearly that requires the participation of the regime, with the support of their allies, and we're hoping Russian Federation will choose to play a constructive role in supporting ceasefires... ultimately through Geneva.
"It is the right moment to talk about this, how the worldwide community, with Russia, Iran, Saudi Arabia, Europe, with the USA, can drive forward a peace process for Syria and avoid further military escalation of the conflict".
"Congress needs to work with the president to try and deal with this long-term strategy, lack of strategy, really, in Syria", he said.
Moscow's concern is that the U.S.
Tensions between the United States and Russian Federation will not "spiral out of control" following last week's US cruise missile strikes on a Syrian air base, Defense Secretary Jim Mattis said on Tuesday, describing it as a one-off response to Syria's use of banned chemical weapons.
Safe Haven Demand Pressures Dollar
China and South Korea agreed on Monday to tougher sanctions on North Korea if it carries out nuclear or long-range missile tests. US crude was little changed at $53.09 a barrel, lingering near a five-week high touched earlier in the session.
On Monday, Prime Minster Justin Trudeau had spoken somewhat less directly about Mr.al-Assad's future, saying: "We need to move as quickly as possible towards peace and stability in Syria that does not involve Bashar al-Assad".
"If we can achieve ceasefires in zones of stabilization in Syria, then I believe - we hope we will have the conditions to begin a useful political process", Tillerson said.
It is our policy for a unified Syria that is governed by the people of Syria.
Tillerson spoke after a meeting of the "likeminded" countries was hastily arranged on the sidelines of the summit of the Group of Seven industrialized economies in Italy.
That likelihood was backed up by White House spokesman Sean Spicer on Monday, who warned Syria against further use of chemical weapons or chlorine-laden barrel bombs.
She declined to say whether Canada would support further unilateral US strikes against Syria in response to the Assad government's use of barrel bombs - long criticized for the damage they inflict on civilians - within its own borders.
The Foreign Secretary pulled out of a visit to Moscow just hours before he was due to fly out, attacking Vladimir Putin's continued support of Bashar Assad's regime.
Tillerson is likely to be grilled by his G7 counterparts to clearly set out USA policy in Syria.
NPR's Lucian Kim reports that the unfolding drama surrounding the attack and its aftermath has soured Tillerson's visit, which had once borne great expectations from US and Russian diplomats.
Tillerson stopped short of accusing the Russians of complicity in a suspected sarin nerve gas attack April 4 that killed at least 87 civilians in Syria's southern Idlib province.
The White House warns Syria that further use of chemical weapons or barrel bombs could bring U.S. military retaliation.
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