Normality and integrality of schemes and splitting of map from structure sheaf to derived-pushforward of structure sheaf along proper birational map

Question

Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, and there exists an $R$-linear map $\phi: S\to R$ whose restriction on $R$ is the identity map, then it is standard that $R$ is also a normal domain. This leads me to ask the following question:

Let $X, Y$ be Noetherian schemes such that $X$ is an integral and normal, and there exists a proper birational morphism $f: X\to Y$. If the natural map $\mathcal O_Y \to \mathbf Rf_*\mathcal O_X$ is a split monomorphism, then is $Y$ also a normal integral scheme?