No, it fails even in the affine case. Let $R$ be a rank-1 valuation ring whose nonzero maximal ideal $m$ satisfies $m^2 = m$ (e.g., the valuation ring of an algebraic closure or completed algebraic closure of a complete discretely-valued field). Let $\pi \in m$ be a nonzero element of the maximal idea, so the fraction field $K$ of $R$ is equal to $R[1/\pi]$. Let $R' = K \times R/m$. Then the natural map $f: {\rm{Spec}}(R') \rightarrow {\rm{Spec}}(R)$ is a counterexample in the affine case because $R \rightarrow R/m$ is formally etale.

What underlies this is the fact that if $A$ is a ring and $J$ is an ideal such that $J^2 = J$ then $A \rightarrow A/J$ is formally etale (that being a non-flat map when $J \ne 0$, so perhaps slightly surprising at first sight, but not really surprising if one reflects on the logical structure of things). This is the standard counterexample to EGA 0$_{\rm{IV}}$ 19.10.3(i) and to EGA IV$_4$ 18.4.6(i) (whose "proof" ends by invoking EGA 0$_{\rm{IV}}$, 19.10.3(i)).

[I originally gave what I thought to be a counterexample in the affine case, but I realized it violates universal schematic dominance, so below I give a non-qc counterexample that was originally a comment.]

Let $A = \mathbf{F}_2^I$ be a product of copies of $\mathbf{F}_2$ indexed by an infinite set $I$, and let $S = {\rm{Spec}}(A)$. Observe that every local ring on $S$ is $\mathbf{F}_2$ since every element of $A$ is idempotent. Let $X$ be the disjoint union of the evident collection of clopen points (indexed by $I$) and the reduced structure $Y$ on the closed complement of their union. Let $f:X \rightarrow S$ be the natural map. This is a universal bijection (built from the stratification by $Y$ and $S-Y$) and an isomorphism on local rings, so faithfully flat. In particular, it is universally schematically dominant. It is not even qc (since $I$ is infinite), so not an isomorphism. The interesting thing is that it is formally etale. This amounts to showing $f|_Y:Y \rightarrow S$ is formally etale.

More generally, if $A$ is any ring and $J$ is an ideal such that $J^2 = J$ (such as $A$ as above and $J$ the ideal of $Y$ in $S$) then $A \rightarrow A/J$ is formally etale (that being a non-flat map when $J \ne 0$ and ${\rm{Spec}}(A/J)$ is not open in Spec($A$), such as happens above, so perhaps slightly surprising at first sight). This is the standard counterexample to EGA 0$_{\rm{IV}}$ 19.10.3(i) and to EGA IV$_4$ 18.4.6(i) (whose "proof" ends by invoking EGA 0$_{\rm{IV}}$, 19.10.3(i)).