Not a chance. Putin is a pragmatist: There is no up side for him to retaliate in that manner, or start firing off nukes, and we have no need to do so if he does not.
Now, will the Russians try to shoot down incoming missiles? Quite possibly, yes. They may have limited, but only limited, success.
Very often, in the course of a street brawl that started over nothing, one or more of the participants end up dead. Putin said he’d retaliate. As a pragmatist, he knows that if he does not retaliate to a strike, he will lose his creds with the Arabs. If Trump doesn’t strike, he loses his creds with everybody. But neither of them want to blow up the world, especially not by accident in a stupid brawl.
So, the best outcome would be for Putin to say he did an extensive investigation, and discovered that this dumbass, Ahmad ibn Kulub Abu Khatoul and his co-conspirator Igor Sonabitchovicz were the ones who launched a gas attack on the village, it was unauthorized and illegal, and we turn them over to the ICC for prosecu—oops. An angry mob just ripped them apart. Matter resolved. Trump stands down, Putin stands down, and nobody loses any creds with anybody.
I’ll bet they’ve already planned something like this in advance.