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Since Jupiter and Saturn's moons are composed of a rock+ice mix. For example, Callisto is 50% rock and 50% ice. When the Sun finally goes red giant, could it melt a significant portion of the 50% of Callisto that is ice? And over what timescale?

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The Sun's luminosity is going to increase by a factor of a thousand or more, so objects at 30 or so AU will receive as much illumination as Earth does now. That is right around the perihelion of Pluto, so I think ice moons will be fairly literally toast. That's even assuming that the swelling is sufficiently slow and spherically symmetric, and the significant mass loss that occurs is such that their orbits aren't disrupted, which might be generous. The entire red giant phase only lasts a few million years, so the transition between phases is instantaneous on astronomical timescales. I'm not sure about human timescale, though. If there are any sentient, corporeal beings living on Earth in five billion years, they better monitor the Sun at all times.

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Jupiter may expand, but I'm not really quite sure about it's going to vaporize, there wil be some meteorological in Jupiter, and in the moons some ice is going to be evaporated indeed, but there are going to be some ice around it..

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