A theorem of Jung states that, given n pairwise distinct points in the Euclidean plane E, there is a unique circle of smallest radius in E that contains all the points and its radius is less than or equal to d/(3^(0.5)) where d is the maximum distance between any pair of the points. Is there an analogous theorem that applies to the 2dimensional surface of a sphere in 3dimensional Euclidean space?

Triangle with side $d$ is the worst example for $d< \arccos(\tfrac13)$. In this case you also have uniqueness. For $d=\arccos(\tfrac13)$, the four point set with pairwise distance $\arccos(\tfrac13)$ is the worse; no uniqueness and $r=d$. It is not clear for me what happens for $d>\arccos(\tfrac13)$. I guess $r=d$, but I do not see the worse configuration except for particular values of $d$. For example, concentric circles with radii $$2\alpha, 4\alpha, \dots, 2k\alpha$$ give the worse configuration for $\alpha=\pi/(2k+1)$, $k$ is a positive integer and $d=\pi\alpha$. 

