You are confusing two issues here, the mathematical format of the force, and the magnitude/strength of the force acting on a body.
The format of the force felt by a pole of a magnet does fall as $1/r^2$, as seen here, which is an approximation, as magnetic monopoles do not exist in nature the way electric ones do.
$$
F=\frac{\mu \, q_{\text{m}1} \, q_{\text{m}1}}{4 \pi r^2}
$$
There are constants and functions entering the problem if one wants to compute relative strengths acting on a body.
F is force (SI unit: newton)
$q_{m1}$ and $q_{m2}$ are the magnitudes of magnetic poles (SI unit: ampere-meter)
$μ$ is the permeability of the intervening medium (SI unit: tesla meter per ampere, henry per meter or newton per ampere squared)
r is the separation (SI unit: meter).
The gravitational force which also falls as $1/r^2$ has different constants in front of it and depends on the masses.
The earth is a huge ensemble of masses and at the distances you describe the gravitational force is constant,$$
F=mg
\,,$$ where $m$ is the mass of the particle and $g$ the gravitational acceleration.
The force generated by the magnetic field has to be computed for the particular situation, but the constants are such that the magnetic force cannot overcome the gravitational force. It is the magnitude of forces that has to be compared for induced motion.