Complemented ideals of $\ell_\infty$

Answering questions raised in \cite{Leonetti, Uzcategui} we characterize ideals $\mathcal I\subseteq \mathcal P(\omega)$ such that $c_{0,\mathcal I}$ is complemented in $\ell_\infty$ as exactly those ideals for which the space $K_{\mathcal I}= \mathsf{Stone}(\mathcal P(\omega)/\mathcal I)$ is approximable, i.e., the unit ball of the space $M(K_{\mathcal I})$ of signed Radon measures on $K_\mathcal I$ is separable in the weak* topology.