Why are the natural satelites (moons) of all planets - including the moons of the gas giants - solid or rocky, and not gaseous?
To answer this, we have to consider the definition of an atmosphere. A popular way of looking at it is to think of an atmosphere as a layer of gases surrounding a body. by that definition, we can say that gas giants are really just planets that are massive enough to accrete substantially large atmospheres (because deep down, they have a rocky core - although not a core similar to the "rocky" inner planets in our solar system). So now we can reduce the question to "Why do most natural satellites not have extremely large atmospheres?"
This comes down to mass. The more massive an object is, the more gravitational pull it has on objects around it. This means that a very massive object traveling through a region of gas would attract the gas more than a less massive object would. In the early Solar System, the more massive bodies far enough out in the protoplanetary disk managed to gain a substantial amount of matter via core accretion, as gas and dust were collected by a rocky core. Moons are unable to do this because they are very low-mass; therefore, they cannot become gas giants (in fact, many moons around giant planets may be the remains of the material the planets never accreted).
Gas dwarfs, low-mass gas planets with small envelopes, are certainly possible. However, they would have to have radii at least 1.5-2 times the radius of the Earth, and masses scaled appropriately. Moons are nowhere near that size.