Godzilla was in many ways originally a metaphor for the enormous mindless destructive capability of nuclear weapons, so you're close. Later films confuse the matter somewhat, but in the original his nuclear origin is clear and in fact the loss of a fishing vessel that starts the film is an echo of a real life incident involving a civilian Japanese vessel (the Lucky Dragon 5) that was exposed to radiation and fallout by accident after an American nuclear test. There is further evidence of the intention - Godzilla's skin was modeled after keloid scars, and of course the destruction caused by Godzilla are directly compared to the atomic bombings in the film itself. Plus, well, there's a lot of interviews out there with Ishiro Honda and others where they spell it out.

Furthermore, the treatment of the "Oxygen Destroyer" device in the first film is something of an echo of the attitudes of some scientists who worked on the creation of nuclear weapons - what was once a matter of great theoretical interest becomes a subject of horror once the destructive capabilities become obvious, and much like Pandora's Box, there is a wish that it had never been created and could be un-created. The Oxygen Destroyer is actually somewhat un-created by the end of the film, but the theme that someone somewhere will build and use another one is also made clear.

It might also interest you that in the most recent Japanese version, Shin Godzilla, Big G very consciously echoes the tsunami and aftereffects of the Fukushima incident - at first a mindless wave of destruction that ends up leaving a large trail of nuclear pollution, with a government bureaucracy struggling to respond quickly.

The original film, made in 1954, was understated about attitudes towards America specifically, as the occupation had ended two years previously in 1952. Direct criticism would have been very controversial for a commercial film at that time, so Americans are not really mentioned at all - the focus is on the destruction and aftermath and the pain related to it. Again for comparison, Shin Godzilla spends a great deal of its time dealing with modern Japan's complex and conflicted relationship with the US and the military situation that entails.

edited to expand somewhat.

edited (again) to reword the bit about Lucky Dragon 5, credit to /u/cckerberos below for calling out my bad wording in the initial.