Not completely. A denser than Earth substance would allow one to get closer to the gravitational center so that less mass would be required to obtain 1 g. As it happens if one could stand on the core of the Earth one would experience 1 g. (The massive core still being to much to launch.)
Read the fine print, dude...I didn't write "as much as the Earth weighs," I wrote "as much as a planet weighs." Even accounting for a high-density object permitting closer proximity, you'd still have at least a Mercury's worth of mass to lift off the ground.
I'm no rocket scientist, so this is just a low-informed stab at the issue. Dense mass seems to me to be the expensive way to solve the problem. We've all seen artificial gravity from spinning. Motion is the cheaper solution, IMHO. You could combine the two, but in the end, it should depend more on motion and less on mass. Enough energy for that motion is practically free in space, with unfiltered sunlight for solar collecters.