A "grave breach" of the Fourth Geneva convention is extensive destruction not justified by military necessity. I'm sure there were militarily significant factories and ports in both cities, but the level of death and destruction seems much more than needed. Or is it justified by the effect on Japanese national morale?
I'm asking this in terms of international law. I'm certain the ethical debate will continue indefinitely.
The Geneva Convention wasn't enacted until 1949, four years after the war ended.
The bombings of cities was common place in all wars up to World War II, because factories and communications where important targets at the time and they were mostly in the middle of cities.
It's been stated that the firebombings of Tokyo where likely more destructive than the A-Bombs in Hiroshima and Nagasaki... Therefor the damage at the time, while enourmous, was fairly standard given what had already been happening in WWII, what made them special it was that these were done by a single bomb instead of massive carpet bombing.
So it's hard to argue that it was a breach of the Geneva Convention since 1) it didn't existed, 2) it was fairly standard destruction given the nature of that war.
If the bombings happened today (or the fire bombings of Tokyo/ Dresden, etc...), they certainly would be considered a war crime. At the time, the US was going by the Hague Convention of 1907 which stated that only the bombing of undefended cities is prohibited. Since Hiroshima and Nagasaki were both defended cities, they were considered fair game.
Going really meta here: a (well-sourced) /r/badhistory post about bad history on /r/badhistory is relevant here, as is the comment there by /u/loony636.
To summarize:
Although there was of course no specific law on nuclear weapons and the most famous law on war crimes did not yet exist, there were nonetheless laws of war at the time which can reasonably be interpreted to prohibit the atomic bombings--in addition to prohibiting many other bombings of civilian sites during WWII.