Let $\Sigma$ be a closed, connected and oriented surface embedded in $\mathbb{S}^3$. Denote by $\overline{K}$ and $K$ the sectional curvatures of $\mathbb{S}^3$ and $\Sigma$, respectively. Then, Gauss' equation gives that
$$K - \overline{K} = K - 1 = K_G,$$
where $K_G$ is the Gauss-Kronecker curvature of $\Sigma$. From Gauss-Bonnet theorem, it follows that
$$\int_{\Sigma} K_G \, \operatorname{dvol}_{\Sigma} = \int_{\Sigma} (K - 1) \, \operatorname{dvol}_{\Sigma} = 2 \pi \chi(\Sigma) - \operatorname{vol}(\Sigma). $$
Thus,
$$\operatorname{vol}(\Sigma) = 2 \pi \chi(\Sigma) - \int_{\Sigma} K_G \, \operatorname{dvol}_{\Sigma}.$$
In particular, if $\Sigma$ is totally geodesic, then $K_G \equiv 0$ and $\chi(\Sigma)$ must be positive. The only option is that $\chi(\Sigma) = 2$ holds, so that every totally geodesic surface of $\mathbb{S}^3$ must be a $2$-sphere.
Is this reasoning correct?