  Let $\Omega$ be homogeneous of degree zero, have mean value zero and
integrable on the unit sphere, and $\mathcal{M}_{\Omega}$ be the
higher-dimensional Marcinkiewicz integral associated with $\Omega$. In this
paper, the author considers the complete continuity on weighted
$L^p(\mathbb{R}^n)$ spaces with $A_p(\mathbb{R}^n)$ weights, weighted Morrey
spaces with $A_p(\mathbb{R}^n)$ weights, for the commutator generated by ${\rm
CMO}(\mathbb{R}^n)$ functions and $\mathcal{M}_{\Omega}$ when $\Omega$
satisfies certain size conditions.
