By stellar remnant, it sounds like you mean a white dwarf. These each have a composition that is determined by their history and how far into nuclear burning they've gone. Often they have lots of carbon and oxygen, sometimes they get as far as iron. But the Earth formed from dust orbiting the Sun, and since its formation mechanism was so different, it has a very different composition.
All the same, it should perhaps be noted that a planet whose metallic core has cooled to a solid is not so much different from a white dwarf. The main difference is the mass is lower, so the kinetic energy of the free electrons is lower, so many of them get captured by the nuclei. So it is like a white dwarf with many fewer free degenerate electrons. That's mostly what the electrostatic attractions are doing-- removing degenerate free electrons. So it has a smaller radius for its gravity, since there are fewer electrons producing the degeneracy pressure. In condensed matter lingo, that population is called the "conduction band."