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.


$Edited$: Take $R=\mathbb{Z}+X\mathbb{Q}[X]$ instead of $\mathbb{Q}[X]$, then $2$ is nonunit in $R$  and $2$ divides $X$ in $R$. Hence $\gcd_R(2,X)=2$.