Sir Michael, writing in the Sunday Times, said the Kremlin was responsible "by proxy" as the "principal backer" of President Bashar al-Assad's regime.
It comes after Foreign Secretary Boris Johnson cancelled a visit to meet his opposite number in Moscow.
The attack has been widely blamed on the Syrian government.
Sir Michael said last Tuesday's gas attack was "barbaric, immoral and illegal" and the response of air strikes from US President Donald Trump was "the right call".
Referring to Russia, Sir Michael wrote: "This latest war crime happened on their watch.
"In the past few years, they have had every opportunity to pull levers and stop this civil war. Russia must show the resolve necessary to bring this regime to heel."
The defence secretary said Syria needed a government in which Mr Assad played no part. He acknowledged achieving that was "not easy, but not impossible".
"By sending Tomahawk missiles to attack the airfield, aeroplanes and equipment believed to be involved, it has sent a strong signal to the Syrian regime to think twice before using gas in future," said Sir Michael.
"Justified and appropriate though US action was, it was also necessarily limited. We now need a long-term solution to this conflict."
Sir Michael said Mr Assad must depart and the search for stability begin.
"Someone who uses barrel bombs and chemicals to kill his own people simply cannot be the future leader of Syria," he stated.
Shadow Foreign Secretary Emily Thornberry said Labour's official position was that the Trump administration should not have acted without UN backing but she conceded that senior figures, including deputy leader Tom Watson, took a different view.