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

Weighted complete continuity for the commutator of Marcinkiewicz integral