I started writing this last night, but didn't finish. In addition to Jason Starr's answer, you can use my new vanishing theorems to remove the assumption that $R$ is essentially of finite type over a field. Note that the special case when $R$ is essentially of finite type over a field of characteristic zero also follows from Kempf's criterion for rational singularities [Kempf 1973, p. 50].

In short, if $Y$ is assumed to be regular, then $R$ will be Cohen–Macaulay. In fact, we have the following stronger result.

**Theorem.** *Let $R$ be a Noetherian ring and let $f\colon Y \to \operatorname{Spec}(R)$ be a proper birational morphism such that $Y$ is regular and $R^if_*\mathcal{O}_Y = 0$ for all $i > 0$. Suppose $R$ contains $\mathbf{Q}$. Then, every localization of $R$ is pseudo-rational, and in particular $R$ is Cohen–Macaulay.*

*Proof.* Since the conclusion can be checked locally, it suffices to consider the case when $(R,\mathfrak{m})$ is local. Set $\hat{Y} := Y \times_{\operatorname{Spec}(R)} \operatorname{Spec}(\hat{R})$, and consider the Cartesian diagram
$$\require{AMScd}\begin{CD}
\hat{Y} @>>> Y\\
@V\hat{f}VV @VVfV\\
\operatorname{Spec}(\hat{R}) @>>> \operatorname{Spec}(R)
\end{CD}$$
By base change, $\hat{f}$ is proper, and by flat base change, $\hat{f}$ is birational [EGAI$_\text{new}$, Proposition 3.9.9] and $R^i\hat{f}_*\mathcal{O}_{\hat{Y}} = 0$ for all $i > 0$. Note that $\hat{R}$ is normal by [Lipman 1969, Remark 16.2] and that the pseudo-rationality of $\hat{R}$ would imply that $R$ is pseudo-rational [Murayama 2022, Proposition 4.20]. Since $\hat{Y}$ is regular by [EGAIV$_2$, Lemme 7.9.3.1], we may replace $R$ by $\hat{R}$ and $Y$ by $\hat{Y}$ to assume that $R$ is complete local.

We now prove the special case when $(R,\mathfrak{m})$ is complete local. First, by my version of Grauert-Riemenschneider vanishing [Murayama, Theorem B(i)], we know that $R$ has rational singularities (see, e.g., [Kollár 2013, Definition 2.76] for the definition). Since $R$ having rational singularities is equivalent to saying that $R$ is pseudo-rational in our setting [Murayama, Remark 7.4], we are done.

We can also show that $R$ is Cohen–Macaulay directly. Since Cohen–Macaulayness can be checked locally, it suffices to consider the case when $(R,\mathfrak{m})$ is local. Since $R$ is normal, the map $R \overset{\sim}{\to} f_*\mathcal{O}_Y$ is an isomorphism [EGAIII$_1$, Corollaire 4.3.12]. Combined with the assumption that $R^if_*\mathcal{O}_Y = 0$ for all $i > 0$, we have the quasi-isomorphism $R \overset{\sim}{\to} \mathbf{R}f_*\mathcal{O}_Y$.
Applying $\mathbf{R}\Gamma_{\mathfrak{m}}(-)$, we obtain
$$\mathbf{R}\Gamma_{\mathfrak{m}}(R) \overset{\sim}{\longrightarrow} \mathbf{R}\Gamma_{\mathfrak{m}}(\mathbf{R}f_*\mathcal{O}_Y) \cong \mathbf{R}\Gamma_{f^{-1}(\{\mathfrak{m}\})}(\mathcal{O}_Y).$$
Now taking $i$-th cohomology modules, we obtain the isomorphisms
$$H^i_{\mathfrak{m}}(R) \overset{\sim}{\longrightarrow} H^i_{\mathfrak{m}}(\mathbf{R}f_*\mathcal{O}_Y) \cong H^i_{f^{-1}(\{\mathfrak{m}\})}(\mathcal{O}_Y).$$
Since the local cohomology modules on the right vanish for all $i < \dim(R)$ by my version of Grauert–Riemenschneider vanishing (in its dual formulation) [Murayama, Theorem B*(i)], we see that $R$ is Cohen–Macaulay. $\blacksquare$

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