I doubt anyone in the I.C.C. thinks he will be easily arrested and taken to the Hague. Russia certainly won't extradite him.
Although that has happened to other rulers, Putin is in a far stronger position than they were. So it might be more symbolic than becoming reality. However it may be easier to take in some of his military commanders, against whom there are already long lists of charges, and we can only hope they are.
I don't know how the charge has been reported to the Russian people, if at all. It might be withheld as much as possible; with what does leak through being dismissed officially as "rumours" or "NATO propaganda" or the like.
We know Moscow has already reacted with utter and rather offensive contempt, to the outside world, but how does a dictator tell his subjects he's been charged with war-crimes when he has banned calling it a "war" and convinced most of them that he is only staging a "special military operation" to (among other claims) "protect the motherland"? It may make his hold over them rather shakier.
The bigger question is what might happen after the war, depending on its outcome. If he loses power, will it create a very dangerous power vacuum? Or have he and his closest top brass already planned the eventual hand-over? Would his replacement be even worse? Would the Wagner Group stage a coup? If I were Vladimir Putin I would be very wary of them, and if they take over, might that be a case of God help the ordinary Russians?
It is notable that Russia, like China, has never known democracy. It's had brief glimpses of reform, under Tsar Nicholas II then after the USSR collapsed, but neither lasted long.