Since nuclear bombs can level entire cities, is it possible that one can destroy a Star Destroyer?
Unfortunately, we have to go way outside of canon to be able to answer this:
The reference Slave Ship and Iron Fist, old EU/Legends books. One thing they mention is that proton torpedoes have a range of explosive yields from "megatons to gigatons" - and that:
The recoil dissipation bracings on Imperial Star Destroyers can withstand "explosions measured in the giga-tonnage range" (ref. Slave Ship)
the Rebel fighters in Iron Fist used their proton torpedoes to blast through the hull armour of an unshielded Executor-class Star Destroyer
This indicates that the Star Destroyer mostly survived, but with a breached hull.
As has been noted - the largest nuclear weapon on earth (so far) was the Tsar Bomba at 50 megatons. This is way down the bottom end of proton torpedo range - and would probably not do too much damage to the hardened armor hulls of a Star Destroyer.
There is also the physics of nuclear explosions in space - http://history.nasa.gov/conghand/nuclear.htm
If a nuclear weapon is exploded in a vacuum-i. e., in space-the complexion of weapon effects changes drastically:
First, in the absence of an atmosphere, blast disappears completely.
Second, thermal radiation, as usually defined, also disappears. There is no longer any air for the blast wave to heat and much higher frequency radiation is emitted from the weapon itself.
So, unless the weapon was exploded directly on the ship surface, there would be almost no blast or thermal radiation. There, would, however be a lot of high energy radiation that would probably weaken hull structure and even kill crew inside the ship - but would not destroy the ship outright.
If the weapon was exploded in contact with the ship's hull, there'd still be an attenuation of the damage - but there would be some blast and thermal radiation into the hull. I'd say that the ship would still survive (again, in EU/Legends, the ships do survive this kind of punishment from multiple torpedo hits).
The final scenario would be an explosion inside the hull - this would maximise the blast and thermal radiation. However, the effect on the ship would be determined by the actual yield of the device and the internal structure of the ship. Chances are though, the ship would not survive as internal structure would fail, and then secondary explosions from weapon stores and the power reactor would pretty much rip the thing apart from the inside.
This is what happened in Star Wars: X-Wing (http://starwars.wikia.com/wiki/Gafra)
the warhead was then delivered to the Invincible in a shuttle modified to appear to be an Imperial vessel, and the Star Destroyer was blown up from the inside.
According to Wookieepedia, a single nuclear weapon was sufficient to destroy a Star Destroyer on at least one occasion:
An Imperial nuclear warhead was captured by the Rebel Alliance from the freighter Gafra during Operation Strike Fear. The warhead was then smuggled onto the Star Destroyer Invincible, destroying it from the inside.
This probably isn't canon but I think it's the best we can get.
This leaves it unclear whether a nuclear bomb would destroy a Star Destroyer it were detonated outside of its hull, or what effect the ship's shields would have. One could try to reason about this case, but empirically, I don't think it can resolved using existing media. Nuclear weapons are not common in the Star Wars universe.
Starship and Space Stations are generally not the most structurally sound things to have ever been constructed to begin with, because they don't need to be. The thing that stops them from just being shot full of holes in Sci-fi is magic shields of one type or another, but this is important to remember that these constructs aren't somehow magically going to withstand the impact of things that structure normally wouldn't. So yes, it can be destroyed with a Nuke.
But here's the thing. Nukes aren't something with such an easily pinned down destructive capacity. Hypothetically, you can blow up a galaxy with a nuke or you can have a relatively small explosive that takes out only something the size the standard car. So when you say "Nuke" that isn't really something that can be categorically stated one way or the other. Some nukes can, some nukes can't.
It is fully possible that a Star destroyer could stand up to, say, the Hiroshima and Nagasaki bombs, but not be able to stand up to anything that we regularly make now.