In terms of natural gravity (the mass of the actual physical structure of the Death Star and everything on/in it, without taking into consideration any technological devices that create artificial gravity), certainly not.
The Death Star isn't even the size of a small moon in the sense that most of us think of moons (rocky objects of roughly spherical shape), by the standards of our solar system. The Death Star II was only about 100/160 kilometers miles in diameter, and the first one was even smaller. As you can see in this list of Solar System objects by size, the only objects with a diameter of 160 kilometers are asteroids, not moons (this list uses radius, which is half of the diameter of the object, so we're looking for objects with a radius between 50 and 80 kilometers). None of these objects have an atmosphere.
A moon that size wouldn't have enough mass to form a sphere, let alone build up and hold on to an atmosphere. Rocky objects that size are lumpy and randomly shaped, often referred to as "potato shaped", for reasons that I don't fully understand. And these objects are, of course, more or less solid rock, with little or no negative space inside, and relatively few cavities.
The Death Star, on the other hand, is full of negative space. It is mostly hollow, at least relative to a moon, because it has to be: it is a space station manned by hundreds of thousands of men and droids, and has living quarters, recreation areas, maintenance areas, utility areas, storage, command facilities, battle stations, etc, etc, etc.
Without artificial gravity of some kind, a mostly hollow structure like this would be incapable of holding on to an atmosphere on the surface - it simply doesn't have enough mass. If, however, it was airtight (which it presumably was) it would have to contain an artificial atmosphere, since we never see anyone inside wearing oxygen masks.
This does indeed seem to be the case. When the Star Destroyer crashes into it, we see what looks like violent outgassing (at least, that's what I think is going on), suggesting that there is little or no atmosphere on the surface, and the pressure inside is much higher than the near-vacuum of space outside.
However, this is science fiction, not the present day state of technology. We couldn't build something like the Death Star, and we can't create artificial gravity, but the people in the Galaxy Far, Far Away can do both of these things. So really, all bets are off.
The first Death Star was completed when it was destroyed, but the second one was apparently less than half finished. It was bigger than the first, but probably had less mass at the time it was destroyed, because most of the living areas and non-essential facilities were not built yet. It seemed to have only a skeleton crew, so to speak - enough men to man the battle stations and do the construction work, but not much more than that. Less men means less space needs to be airtight and filled with breathable air. Less of the outer shell being in place means the area that could be pressurized is much smaller.
One might argue that the construction teams needed air while they were building the damn thing, but I would imagine that much of the work on the outer portions of the station was performed by droids.
But in any case, since this is science fiction, they can basically do whatever the writers want or need them to do. So is it possible that the exterior of the Death Star had an atmosphere? Of course. But if it did, it was because of some sort of artificial gravity technology, not because of the natural mass of the structure itself. And I'm not sure why they would need an atmosphere outside the shell of the Death Star - I don't think anyone goes out there very often. All the good stuff is inside.
On the scientific side of the question, I found an interesting article about the conditions necessary for an astronomical object to maintain an atmosphere, as explained by a member of the Mars Pathfinder team.