That doesn't seem too plausible.
The moon is only about 1% of the mass of the earth, so let's start with a baseline estimate that it could change surface gravity by 1%. However, it's pretty far from earth's surface (around 100 times earth's radius), and gravity scales with the square of the distance, so now we're talking a 0.0001% change.
It gets worse, though: because we are in mutual orbit with the moon, we are actually in microgravity (freefall) relative to it. In other words, the moon pulls on you with roughly the same strength as it pulls on the earth, so the total 0.0001% change in gravity mostly cancels out! The only part of it that's left is from you being slightly closer to the moon than the earth is (when the moon is above you) or slightly farther (when it's on the other side of the earth). This difference is what causes the tides, but it's incredibly small relative to the gravitational pull of the earth.
Not to mention that the moon was probably not around when the earth was fully molten, or that the earth *does* have heavier elements (iron) predominantly in the center, or any number of other reasons why this idea doesn't seem likely...