There are several considerations:

1. The International Criminal Court did not exist prior to 2002 (The Rome Statute). Prosecution of "war crimes" was extremely limited prior to that (and you could say, still are). Mostly it was the Nuremberg trials that people know about.
2. "Might makes right".

You seem to be coming at the question from an assumption that any targeting of civilians was a war crime. There are thousands of examples throughout modern history of the intentional targeting of civilians by military forces prior to the creation of the ICC, almost all of which went without being prosecuted. The scale of the firebombing in Germany and Japan was obviously unprecedented, but the principal was no different than the Blitz, the Japanese invasion of China, and all the other brutal acts of WWII and wars before it (recall The March to the Sea).

While the Geneva Conventions of the 19th and 20th Century outlined some basic doctrine of what would constitute a "War Crime", there was no active prosecution because there was no formal system set up to do so. Even under the GC, attacks from planes were not prohibited- and it worked both ways, Germans and Japanese likewise were not prosecuted for having attacked civilian populations by air.

Curtis Le May was the general that spearheaded the switch to nighttime low altitude firebombing. He has an interesting quote on the matter:

Killing Japanese didn't bother me very much at that time. … I suppose if I had lost the war, I would have been tried as a war criminal. Fortunately we were on the winning side. Every soldier thinks something of the moral aspects of what he is doing. But all war is immoral and if you let that bother you, you're not a good soldier.

You can interpret this yourself, but I believe he was stating the political reality that winning the war made you relatively safe from prosecution. That being said, the postwar trials did generally excuse Axis commanders that had done nearly exactly what the Allies had done (particularly in the area of unrestricted submarine warfare).

The Vatican is one of a small number of nations that has had a consistent policy on the subject of aerial bombardment, having opposed it since its invention based on the concepts of discrimination and proportionality found in Just War Theory. To be fair though, the Vatican has also not had access to bombers (or an airfield for them to take off from). The papacy was criticized in WWII for its opposition to aerial bombardment by both the Axis and Allies at various times, and was accused of supporting the other side by both sides of the conflict.

The historical background on the legality of bombing civilians is covered in significant detail in Goda, Paul "The Protection of Civilians from Bombardment by Aircraft: The Ineffectiveness of the International Law of War" 33 Mil. L. Rev. 94 1966. The legal answers to your question are there.

Interestingly, the League of Nations unanimously approved a proclamation against bombing civilians in 1938.

PROTECTION OF CIVILIAN POPULATIONS AGAINST BOMBING FROM THE AIR IN CASE OF WAR

^Unanimous ^resolution ^of ^the ^League ^of ^Nations ^Assembly, ^September ^30, ^1938.

^The ^Assembly,

^Considering ^that ^on ^numerous ^occasions ^public ^opinion ^has ^expressed ^through ^the ^most ^authoritative ^channels ^its ^horror ^of ^the ^bombing ^of ^civilian ^populations;

^Considering ^that ^this ^practice, ^for ^which ^there ^is ^no ^military ^necessity ^and ^which, ^as ^experience ^shows, ^only ^causes ^needless ^suffering, ^is ^condemned ^under ^the ^recognised ^principles ^of ^international ^law;

^Considering ^further ^that, ^though ^this ^principle ^ought ^to ^be ^respected ^by ^all ^States ^and ^does ^not ^require ^further ^reaffirmation, ^it ^urgently ^needs ^to ^be ^made ^the ^subject ^of ^regulations ^specially ^adapted ^to ^air ^warfare ^and ^taking ^account ^of ^the ^lessons ^of ^experience;

^Considering ^that ^the ^solution ^of ^this ^problem, ^which ^is ^of ^concern ^to ^all ^States, ^whether ^Members ^of ^the ^League ^of ^Nations ^or ^not, ^calls ^for ^technical ^investigation ^and ^thorough ^consideration;

^Considering ^that ^the ^Bureau ^of ^the ^Conference ^for ^the ^Reduction ^and ^Limitation ^of ^Armaments ^is ^to ^meet ^in ^the ^near ^future ^and ^that ^it ^is ^for ^the ^Bureau ^to ^consider ^practical ^means ^of ^undertaking ^the ^necessary ^work ^under ^conditions ^most ^likely ^to ^lead ^to ^as ^general ^an ^agreement ^as ^possible:

I. ^Recognizes ^the ^following ^principles ^as ^a ^necessary ^basis ^for ^any ^subsequent ^regulations:

1) The intentional bombing of civilian populations is illegal;

2) Objectives aimed at from the air must be legitimate military objectives and must be identifiable;

3) Any attack on legitimate military objectives must  be carried out in such a way that civilian populations in the neighbourhood are not bombed through negligence;

This obviously had no effect on the various parties in WWII; see Goda's article for details.

Also, Dresden

It can be and has been argued[1] that Dresden was a legitimate military target. Its factories, in the heart of the city, made things like ball bearings and optical weapon/bomb sights vital to the German war effort. The nature of Allied bombing technology at the time precluded precision strikes, and the number of German fatalities was greatly compounded by mismanagement of local officials involved in emergency preparedness.

[1]http://www.amazon.ca/Dresden-Tuesday-February-13-1945/dp/0060006773

I asked a somewhat similar question a couple of months back: how did we get from indiscriminate bombing of cities being a widely accepted (but not uncontroversial) tactic to it being widely considered a war crime?

I only got one reply to that, but it does have some interesting detail from a military perspective. Essentially, the poster was of the view that it was changes in technology, allowing for more precision bombing that permitted, if not caused, the shift.

Another important point, that no-one has mentioned yet, is the continued existence of nuclear weapons and the doctrine for their use. That is based on the use of high yield weapons to threaten cities. Implicitly: if there is an existential threat to a nuclear state, they will throw the usual rules out the window.

Intentionally targeting civilians

You're starting from a faulty premise. The US never made it a policy to intentionally target civilians. The US switched to area bombing because most of the larger industrial targets had been flattened and we strongly suspected that the Japanese had dispersed their industry into residential buildings. To quote Curtis Lemay, from his own lips:

No point in slaughtering civilians for the mere sake of slaughter. Of course, there is a pretty thin veneer in Japan, but the veneer was there. It was their system of dispersal of industry...I'll never forget Yokohama. That was what impressed me: drill presses. There they were, like a forest of scorched trees and stumps, growing up throughout that residential area. Flimsy construction all gone...every house burned down, or up, and drill presses standing like skeletons

So, if in fact the Japanese had dispersed their industry into residential areas(and they had, as we verified with post-bombing damage assessment flights) it was a completely reasonable thing to do. Remember, in March 1945(and all the way up through the Summer) the US was still planning on invading the Home Islands in order to bring the war to a close as soon as possible. It would have been beyond criminal, from the viewpoint of the leadership planning the invasion, to not do anything in their power to reduce the amount of war material available to the defenders.

A good book that discusses this(and the Atomic Bombings, Operation Starvation, planning for the invasion, etc) and the reasoning behind the tactics is Downfall: The End of the Imperial Japanese Empire by Richard Frank. LeMay quote is taken from it.

This is a day late (and too short, in any case), but to address two points.

One is that there was no international law prohibiting the targeting of civilians of the time. There were many precedents that indicated that the official stance of the US was to avoid this if possible, but by the late stage in the war the US had given up on all pretext that these were in any way binding. Additionally, from a purely legalistic standpoint, the London/Nuremberg Charter was set up to only prosecute war crimes by Axis nations. Such are the prerogatives of the victor.

The second is a more troublesome question of how we regard these things. I generally push back against those who claim that it was just a natural evolution of tactics. Choices were made and they were surprising to many. The Secretary of War, Henry Stimson, was shocked at LeMay's change in tactics because he was not consulted and Stimson thought it was a form of butchery. But he did not intervene other than to request it be limited and a specific city (Kyoto) be saved.

There were others, especially in the UK, who thought that this kind of tactic was counterproductive and morally repugnant. It is not a present-day, backwards-looking sentiment to feel this way.

In the US, at least, we have tended to regard World War II as the paradigmatic "good war." And, indeed, the forces on the Axis side were pretty bad. But in elevating this to nostalgic status, we've perhaps made stronger contrasts in the methods of fighting than we ought to have.

At the 50th anniversary of WWII (1995), the veterans groups really managed to change the dominant US discourse on these kinds of questions and put them into a light that was very hard to challenge without getting a lot of outcry. Even something as relatively innocuous as talking about the victims of these kinds of attacks was seen as an attack on the sanctity of the war and those who fought in it. It produced a very polarized historical discourse of "traditionalists" and "revisionists" that, one might be surprised to find, was not nearly as stark in the years prior. It will be interesting to see how the next, upcoming anniversary (70th) does in this respect, but as a friend of mine remarked last night, the one that will probably be the most interesting will be that of 2025 (80th), which will perhaps have given us all enough distance that the passions over it all will have subsided a bit.