If $M$ is a maximal Cohen-Macaulay module for a Noetherian Gorenstein ring $R$, then it follows easily by induction on the projective dimension of $N$ that $\operatorname{Ext}^i_R(M,N)=0$ for all $i>0$ if $N$ is finitely generated of finite projective dimension.

So if $R$ also has finite global dimension, then $\operatorname{Ext}^i(M,N)=0$ for $i>0$ for all finitely generated modules $N$, and so $M$ is projective.