Take $S=\mathbb{Z}[X]$ and $R=\mathbb{Q}[X]$. Then $gcd_S (2,X)=1$ and $gcd_R(2,X)=2$,
where both $R$ and $S$ are GCD-domains.
$Edit$: Take $R=\mathbb{Z}+X\mathbb{Q}[X]$ instead of $\mathbb{Q}[X]$ , then $2$ is nonuunit in $R$ and $2$ divides $X$ in $R$. Hence $gcd_R(2,X)=2$