Jupiter would need to be about 80X it's mass to be the smallest possible red dwarf, so at 2X the mass we're still very far away from star territory. Oddly enough, though, it would be the roughly the same size as it is now, as Jupiter is about as large in terms of volume as a gas giant (or brown dwarf) gets.

As gas giants get more massive they run into an interesting situation in which the increase in gravity compresses the planet just as much as the planet increases in mass--so even though the mass increases the volume remains roughly the same. It just gets denser.

However, the additional mass likely would have drastically effected the evolution of our solar system. It may have flung the earth out of the solar system, or into the sun. It may have prevented as many water-rich asteroids and comets from hitting the earth, preventing it from ever accumulating enough water to support life.

Simulations would be required to figure out how, but it's safe to say that a Jupiter that is 2X the mass would have resulted in a solar system that looks very different from the one we see today.

It doesn't need to swell into a red giant for the earths oceans boil away. The sun slowly burns hotter and hotter as the proportion of helium to hydrogen increases. When the sun first formed it was something like 75% the brightness, and has slowly increased in brightness.

The earth has another ~500 million years of habitability before the sun gets too hot and boils off the oceans