When faced with a constant target density, geodesic slice sampling on the sphere simplifies to a geodesic random walk. We prove that this random walk is Wasserstein contractive and that its contraction rate stabilizes with increasing dimension instead of deteriorating arbitrarily far. This demonstrates that the performance of geodesic slice sampling on the sphere can be entirely robust against dimension-increases, which had not been known before. Our result is also of interest due to its implications regarding the potential for dimension-independent performance by Gibbsian polar slice sampling, which is an MCMC method on $\mathbb{R}^d$ that implicitly uses geodesic slice sampling on the sphere within its transition mechanism.

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