Actually, it's easier than the general curvature case: Let $X^\flat$ be the $1$-form dual to $X$ via the metric $g$. Then the orthogonal plane field to the integral curves of $X$ is integrable if and only if $$ X^\flat \wedge \mathrm{d} X^\flat = 0. $$

In the case of the $3$-sphere and the vector field $X$ that generates the Hopf fibration, the $3$-form $X^\flat \wedge \mathrm{d} X^\flat$ is nowhere vanishing, so the $2$-plane field is not integrable, so there is no foliation to discuss. In particular, $X^\flat$ defines a contact structure on the $3$-sphere.

Moreover, Arnol'd proved that there is no real-analytic foliation of the $3$-sphere by surfaces. (Reeb constructed a famous smooth foliation of $S^3$, though.)

If you are looking for information about codimension $1$ foliations of manifolds, you should, perhaps, look up references that discuss secondary characteristic classes, such as the Godbillon-Vey class, and their geometric and dynamical meaning.

Actually, it's easier than the general curvature case: Let $X^\flat$ be the $1$-form dual to $X$ via the metric $g$. Then the orthogonal plane field to the integral curves of $X$ is integrable if and only if $$ X^\flat \wedge \mathrm{d} X^\flat = 0. $$

In the case of the $3$-sphere and the vector field $X$ that generates the Hopf fibration, the $3$-form $X^\flat \wedge \mathrm{d} X^\flat$ is nowhere vanishing, so the $2$-plane field is not integrable, so there is no foliation to discuss. In particular, $X^\flat$ defines a contact structure on the $3$-sphere.

Moreover, Haefliger (Comment. Math. Helv. 32 (1958), 249–329) proved that there is no real-analytic foliation of a simply-connected compact $3$-manifold by surfaces. (Reeb famously constructed a smooth foliation of $S^3$, though.)

If you are looking for information about codimension $1$ foliations of manifolds, you should, perhaps, look up references that discuss secondary characteristic classes, such as the Godbillon-Vey class, and their geometric and dynamical meaning. A good place to start would be H. Blaine Lawson's article Foliations (Bulletin of the AMS 80 (1974), 369–418).

Note: In the first version of this answer (written in 2014), I had mistakenly attributed Haefliger's theorem to Arnol'd. My thanks to Dan Asimov for pointing out my error.