We assume $S$ connected and orientable surface.
Proof: Let $k_1=k_2=:\kappa$ be principal curvatures.
From $\kappa(p) I(\partial_u,\partial_u)=-II(\partial_u,\partial_u)$ we see that $\kappa$ is differentiable since $I(\partial_u,\partial_u),II(\partial_u,\partial_u)$ are.
Furthermore $$N_{*p}(\partial_u)=\kappa(p)\partial_u$$
$$N_{*p}(\partial_v)=\kappa(p)\partial_v$$
Differentiating both equations with respect to $v$ and $u$ as in Do Carmo book we get:
$$\partial_v N_{*p}(\partial_u)=\partial_v(\kappa(p)\partial_u)=\partial_v(\kappa(p))\partial_u+\kappa(p)\partial_v\partial_u$$
$$\partial_u N_{*p}(\partial_v)=\partial_u(\kappa(p)\partial_v)=\partial_u(\kappa(p))\partial_v+\kappa(p)\partial_u\partial_v$$
Now subtracting:
$$ \partial_v N_{*p}(\partial_u)-\partial_u N_{*p}(\partial_v)=\partial_v(\kappa(p))\partial_u+\kappa(p)\partial_v\partial_u - \partial_u(\kappa(p)) \partial_v-\kappa(p)\partial_u\partial_v$$
Then by Schwarz's Theorem:
$$ \partial_v N_{*p}(\partial_u)-\partial_u N_{*p}(\partial_v)=\partial_v(\kappa(p))\partial_u- \partial_u(\kappa(p)) \partial_v$$
And here I'm stuck.
Why is left side zero?