Depends on the culture The Japanese viewed siege tactics as cowardly and armies at the gates would deliver food and supplies to the people in the walls. Ba Sing Se was able to convince it's citizens there wasn't even a war going on, I don't think they were starving or being killed with siege weapons.
There's no central authority, but it's not a machiavellian free-for-all like medeival Europe. The rest of the world was rather unhappy with the Fire Nation's aggression, even in light if the world's long history of warfare. He would be tried for that, no doubt.
And his reputation/nickname is subtext for crimes he did commit but that the cartoon couldn't spell out.
No, it's not a war crime that I can find, however we can attribute harm caused to civilians through these actions, such as starvation due to supply lines cut off. So he did some vile shit, had a moment where he realized the error in his ways, then did everything in his power to make things better.
Except there no evidence of starvation in Ba Sing Se. After all, there was so little impact that the citizens could be convinced that there was no war.
As to supply lines, earth benders cannot be locked in by a siege. They can create tunnel networks with a literal wave of the hand.
I mean really the existence of war crimes relies on the existence of treaties between the nations defining what those crimes are. Gonna guess the Fire Nation was not a signatory.
The Nazis weren't signatories to the Nuremberg charter, yet they were judged by it. So there is precedent for judging war crimes without pre-existing law.