I believe in Ant-Man shrinking is accomplished by reducing the space between the subatomic particles, and the spaces between the atoms. The how this works goes unexplained (and I don't believe any theory grounded in physics can begin to explain this), and as I can offer nothing there, I'll brush over it.
Ignoring most of the issues this causes, I'll focus a quick discussion around one topic Orejano brought up, specifically gravity. Earth would maintain its mass, but would now be smaller than the eye can see. Even if we dial it back to being roughly 1 meter cubed, we get a density of 5.972 × 10^24 kg per cubic meter.
We can take a trip over to Physics Stack Exchange to determine the density of a black hole. The answer? Seems to be (at the heavy end) 6 × 10^18 kg per cubic meter.
Meaning even if we scale Earth to one meter cubed and not microscopic we're denser than a black hole.
Not a pretty outcome for the citizens of Earth.