The minimum mass that an object would need to initiate deuterium fusion in its core is about 13-14 times the mass of Jupiter. At that point it would be a brown dwarf star. The minimum mass to initiate hydrogen fusion is about 80x the mass of Jupiter. At 2x the mass, Jupiter is well below both limits and would not become a star.

As for the gravitational effect of doubling the mass of Jupiter on the remaining Solar System bodies - of course it would have an effect. Jupiter was (probably) the first planet to form out of the original Solar nebula. At double the mass, it would have had an influence on a larger area of the nebula, therefore pulling in more of the gasses. This would have a greater effect on the formation of the other gas giants. Exactly what that effect would be is up for you to speculate (or run your own Monte-Carlo models) on.

Jupiter is also cited as a major influence of the movement of the planetesimals which formed the inner (rocky) planets. Doubling the mass would certainly change that influence. Again, feel free to run your own Solar System simulations to figure out how that affects planetary formation, but the end game is that the Solar System would be different.