Why don't any of the gas giants have moons of Earth's mass (or greater) that orbit them? Is this generalizable to exoplanet gas giants? Now, we know that the composition of the outer planets tend to contain a much higher ratio of ice over rock. [1] So the lack of Earth-like moons around the outer planets that this could be an artifact of the fact that in our solar system, all the gas giants are beyond the "ice line". 
But could other factors be involved too? If Jupiter was at the distance of the Earth (probably by migrating in), then is it unlikely for it to have a moon of Earth's mass or greater?
[1] This is probably because the amount of rocky material falls off with distance from the Sun, but this is not true for icy material because icy material will volatilize if it's too close to the Sun.
 A: I don't think your first paragraph around ice and rock has anything to do with the question on whether gas giants should have Earth-sized moons.
I will try to find the two references I got this from years ago, but the core concept is around the Roche limit. When an accretion disk begins to form a rocky moon there is only a small region where it can form and remain stable. Inside this region it will tend to break up due to tidal forces and spiral in or spread, and outside the region it may escape. 
One of the effects of this is the way moons' sizes (in general) tend to follow a bulging curve, small to large to small again. And the possible sizes in the stable region depend on the mass of the primary.
So even if Jupiter could be stable at Earth's orbit, it still wouldn't be likely to have a moon the size of Earth.
It could - as objects can be captured, but it isn't likely to accrete.
A: I think it is possible if you look at Ganymede it is bigger than Mercury but however if you know about other planetary systems there are gas giants much bigger than Jupiter. It is very possible for a gas giant to have a moon similar in size of earth however not in terms of mass. We can speculate that if the gas giant is a brown dwarf than it may be possible because there may be two stars forming in the so and so star system (as an example) so the so the brown dwarf have a lot of matter around it and it can accrete to form similar size to that of earth not in mass terms because there is heavy molten iron & nickel inside the earth's core which makes it more massive and the densest planet in the solar system. I think even higher density per cubic meter than the sun. 
