We theoretically analyze the properties of a geodesic random walk on the Euclidean $d$-sphere. Specifically, we prove that the random walk's transition kernel is Wasserstein contractive with a contraction rate which can be bounded from above independently of the dimension $d$. Our result is of particular interest due to its implications regarding the potential for dimension-independent performance of both geodesic slice sampling on the sphere and Gibbsian polar slice sampling, which are Markov chain Monte Carlo methods for approximate sampling from essentially arbitrary distributions on their respective state spaces.