To answer this, we have to consider the definition of an atmosphere. I'll go with Miriam-Webster's online definition: "A mass of gases that surround a planet or star". By that definition, we can say that gas giants are really just planets that are massive enough that they have really massive atmospheres (incidentally, Wikipedia's article on gas giants seems to agree), 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 cloud of gas would attract the gas more than a less massive object would. In the early solar system, this was the case, and it is why Jupiter and the rest of the gas giants have such large atmospheres compared with the smaller bodies in the solar system - namely, the natural satellites. Moons can't hold a lot of atmosphere (at least, not enough the consider them as gas giants) because they are very low-mass; therefore, they cannot become gas giants.
I hope this helps.
Definition: http://www.merriam-webster.com/dictionary/atmosphere
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