Here is a "physics" "proof": Far from any gravitational forces, the napkin ring sits captured between two parallel rigid glass plates at z=r and z=-r. The height 2r cylindrical center is also made of rigid glass. However the curved outer surface is a flexible membrane and the interior of the napkin ring is filled with the appropriate volume of incompressible water. Even though the membrane is free to move (other than being fixed at the plates)  the water will hold the napkin ring shape (the portion of a sphere between the plates). Why? Imagine extending the membrane to be a full sphere of radius R, filling the previously empty cylinder (and caps) with water and then having the glass disappear. The fluid will hold its spherical shape due to fluid pressure. The portion of water which was in the napkin ring won't know that anything changed, the glass exerted the same forces as the new water does along the same contour. This shows that the shape of the napkin ring is completely determined by the volume of water it encloses and not by R (so long as `R>r`.). Since this volume is independent of R, we might as well use R=r. QED?

Variation: Start with a volume of $4/3 \pi r^3$ of fluid in a membrane. It will naturally take a spherical shape. Imagine a long cylinder of glass (inner radius 0) passing along the axis and glagg plates tangent at the poles. Now let the radius of the centeral cylinder grow. The water always thinks it is part of some bigger volume of water and keeps a spherical shape (since the glass presses back on the water with the force that the water presses on it).