Yet despite the Defence Secretary's strong words today, the Government is continuing to face questions over why Foreign Secretary Boris Johnson cancelled a trip to Moscow at the eleventh hour.
But the regime has denied any responsibility, saying its air strikes hit a rebel depot containing chemical materials.
He said a solution could only come about if Mr Assad quit as president.
Referring to Russia, Sir Michael wrote: "This latest war crime happened on their watch. Russian Federation must show the resolve necessary to bring this regime to heel". He acknowledged achieving that was "not easy, but not impossible".
Fallon also made remarks regarding the attack and said that the Syrian crisis needed a permanent solution. Obama stepped back from military action after the parliament in the United Kingdom, a crucial ally, declined to participate and public support in the US waned. "You train the opposition to go after [Syrian President Bashar] Assad".
Earlier on Sunday, Rep. Adam Schiff of California, the top Democrat on the House Intelligence Committee, said on ABC News' "This Week" that Russian Federation is "complicit" in the chemical attack in Syria.
But Tillerson, who is expected to visit Moscow on Wednesday for talks with Russian officials, said on ABC's This Week program there was "no change" to the US military posture towards Syria.
China urges US, North Korea to maintain calm
Both of these scenarios were part of a North Korea review prepared ahead of Trump's meeting with the Chinese leader. The relationship developed by President Xi and myself I think is outstanding.
The Russian foreign ministry said Mr Johnson's decision not to visit Russia was "absurd".
Scores of people died and many were injured on Tuesday after the regime air strike dropped chemical bombs onto the town of Khan Shaykhun.
US President Donald Trump said that the attack was a response to the alleged chemical weapon use in Syria's Idlib province on Tuesday, which resulted in the death of over 80 people, an incident which Washington blames on the Syrian government.
Vladimir Safronkov, Russia's deputy United Nations ambassador, said before the strikes were made public that any US military action will have "negative consequences".
WATCH: Syria Strike: Finally some action or risky reaction?
Iran's President Hassan Rouhani has said "terrorists are celebrating" the U.S. strikes on the air base.
While Nikki Haley, the US ambassador to the United Nations, described regime change in Syria as a USA priority and inevitable, Tillerson suggested that last week's American airstrikes in retaliation for the chemical attack hadn't really changed USA priorities toward ousting Assad.