By stereographic projection we can assume that the given Jordan arc
lies on a sphere, and that its two ends are at opposite poles, $N$
and $S$. Now project the arc onto a cylinder that touches the sphere
at the equator, so that $N$ and $S$ go to infinity on opposite ends
of the cylinder.

Unrolling the cylinder, we now have our arc on a strip of the plane,
with the ends of the arc at opposite ends of the strip. Now we can
inverse stereographically project the arc back onto the sphere, so
that *both* ends go to the point of projection. We now have a closed
Jordan  curve on the sphere, and we can apply the Schoenflies theorem.