Would it be possible for an earth like planet the size of a gas giant to exist?
It would be very unlikely, IMO. Here is why. The reason that the four inner planets are rocky and terrestrial is that close to the Sun molecules like hydrogen, water, ammonia and methane tend to stay as gases and not produce grains and clumps and so on. And because they stay in gas form, the radiation pressure from the Sun and the solar wind will push them outwards. Only at the distance of the orbit of Jupiter it becomes cold enough for ice to form. The gaseous molecules can produce grains and lumps so there’s a lot of this less dense material around to accrete into planets. So the inner planets accreted from grains containing heavier atoms like oxygen and aluminium and silicon as well as iron and nickel and so on. But the total fraction of that in the Solar system is extremely small, much less that 1% of all material available. And that’s why terrestrial planets are small by necessity. There is just not enough of the heavy, rocky stuff around.
