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You can rephrase your question as follows: Let $U,V$ be random (=Haar distributed) independent unitaries. You can write $G=UD_1 V$ where $D_1$ is diagonal (entries are the singular values of $G$, independent of of $U,V$, and follow the Wishart distribution). Now you ask about the eigenvalues of $(D_1VD^2V^*D_1)^{1/2}$. The limit of the empirical measure can be computed by free probability methods. It is the free multiplicative convolution of the law of $D^2$ and that of $D_1^2$, pushed forward by square-root.