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.
Edit. Ryan Budney has pointed out the flaw in this argument, so I withdraw it as it stands. Thanks for the amendment, Ryan.