It is also true for local rings. Let $A$ be a matrix with coefficients in $K = \mathrm{Frac}(R)$, such that $A \mathrm{GL}_n(R) A^{-1} \subset \mathrm{GL}_n(R)$. Then in particular, $A (I_n + E_{i,j})A^{-1} \in M_n(R)$ and $A (I_n+E_{i,i}+E_{i,j}+E_{j,i})A^{-1} \in M_n(R)$ for all $i \neq j$, so that $a_{i,j} b_{k,l} \in R$ for all $i,j,k,l$, where $a$, $b$ denote the coefficients of $A$, $A^{-1}$.
Since $\sum_j a_{1,j}b_{j,1}=1$, there is a $j_0$ such that $a_{1,j_0} b_{j_0,1} \in R^{\times}$. Let $A'=a_{1,j_0}^{-1} A$. Then the coefficients of $A'^{-1} = a_{1,j_0} B$ are in $R$, and those of $A'$ also because $\frac{a_{i,j}}{a_{1,j_0} }= \frac{a_{i,j}b_{j_0,1}}{a_{1,j_0}b_{j_0,1}} \in R$.
For a general $R$, this just shows that $A \in K^{\times} \mathrm{GL}_n (R_m) $ for any maximal ideal $m$ of $R$.