Nuclear weapons are, to date, humanity's most Earth-shattering weapon. They have more than enough destructive power to wipe out the human race for good. But could we actually use them to destroy the Earth itself?
One way to see this is to compare the energy of a nuclear blast to that of the rotational motion of the Earth. The largest nuclear bombs have an explosive energy of several tens of megatons, or about 10^17 Joules, whereas the Earth's rotational energy is around 10^29 Joules.
Yeah, that's a pretty big difference. Spirber notes that the energy of the largest nuclear blast is less than that released by the 2004 earthquake that caused the tsunami.
The amount of fault-moving ("Earth-slimming") energy in this magnitude 9.3 earthquake was estimated at more than 10&^22 Joules, or roughly 100,000 times that of the biggest nuclear bombs. So any effect of a nuclear blast on Earth's rotation would be far below what is measureable.
So, it seems unlikely that one nuke — or several — of today's technology could do harm to the planet (though the environmental effects are quite a different story).
The tsunami did, however, alter the Earth's rotation.
Scientists calculated that the colossal tsunami-causing 2004 Sumatra earthquake caused a slimming of the Earth that shortened the day by a few millionths of a second and shifted the North Pole by an inch.
Do we have enough nukes to achieve a similar effect?
The researchers at the Arms Control Foundation suggest it's possible. According to their calculations, between Russia and the United States, we have about 26,000 warheads — there are only about a thousand between the rest of the countries that officially, unofficially and probably have nukes. If one assumes — a big assumption, but let's go for the doomsday scenario — that each nuclear weapon stockpiled the world's nuclear leaders has an energy of 10^17 Joules, then between Russia and the United States, we have the potential capacity to release 2.6^22 Joules — or approximate 25% of the energy of the 2004 tsunami-causing earthquake.
So by concentrating the capacity of the world's nuclear arsenal in one place, we might be able to shift the North Pole by one-quarter of an inch. Doesn't exactly seem worth it. But it's for science!