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 $gcd_R(2,X)=2$