Putin is indeed guilty of war crimes, and in a just world would stand trial for them. I would go so far as to say he deserves the death penalty. If you oppose the death penalty, then you should at least conclude he deserves whatever you think is the most severe allowable punishment.
At the very least, Putin is guilty of ordering a blatant war of aggression (the central crime established in the Nuremberg trials). It is highly likely (though admittedly not yet definitively proven) that many of the atrocities committed by Russian troops in Ukraine are at his orders, and not simply the bad behavior of rogue individuals and units.
That said, as a practical matter, it's almost impossible to try and punish Putin so long as he is in power. Successful war crimes trials require either total victory of the kind that ended World War II, or at least some kind of coup or regime change in the nation in question (as with MiIosevic and a number of African dictators and generals charged before the ICC). Ukraine could potentially try lower-ranking Russians they capture. Even if they were acting under orders, that is not a sufficient defense against war crimes (another principle established in post-WWII trials of German and Japanese military men). That might do some good, but nowhere near as much as trying the high-level malefactors who are most responsible.