When we're presented with an Earthly and contemporary film setting, we have no reason but to assume that the laws and status quo also applies to this, unless the film lets us know up front otherwise. It's the reason why Japan nuking Godzilla in Kingdom of Monsters is so damn off-putting. We read the story assuming that this world is exactly like ours (sans the monsters), then BOOM; a country that isn't allowed to have a standing army and is one of the biggest proponents of nuclear disarmament just chucks one wily-nily. You've thrown your audience a curve-ball; this isn't our world anymore, this is a world with a completely different history and set of political dynamics that aren't even remotely explained or anything like ours. If Eric Powell wanted to tell an "Alternate History" story of a militaristic Imperial Japan fighting monsters or something (and skreeonk that would be cool), that is one thing, but since that clearly wasn't his intention we have no reason but to assume it's just bad writing.
It still wouldn't be as weak as the whole blackhole gun shinanigans because that is from an universe of established science-fiction-y goodness.
A nuclear weapon that Godzilla just "shrugs off" is, quite frankly, just juvenile macho-posturing. It's like Batman shrugging off getting shot in the mouth because Batman "is such a skreeonk badass". If there is ever an incarnation of Godzilla then can shrug off an Atom Bomb, then you've basically killed your tension because there isn't anything that can hurt him. Period. I know Godzilla is "invincible" and all that jazz, but even uber-tanks like Heisei Godzilla are stalled and roar when he's hit with a laser beam (also bleeds quite regularly).
If a nuclear weapon is brought into a Godzilla film, it had better damn well be the whole point of the film. It had better be the climax and it had better kill him skreeonk dead. But at that point, we aren't discussing "just nuking Godzilla", we're talking about entire story ideas.
“I have never listened to anyone who criticized my taste in space travel, sideshows or gorillas. When this occurs, I pack up my dinosaurs and leave the room.” - Ray Bradbury