We employ N-body/three-dimensional gasdynamic simulations of the formation of galaxy clusters to determine whether cluster X-ray morpholgies can be used as cosmological constraints. Confirming the analytic expectations of Richstone, Loeb, & Turner (1992), we demonstrate that cluster evolution is sensitive to the cosmological model in which the clusters form. We further show that evolutionary differences are echoed in the gross morphological features of the cluster X-ray emission. We examine current-epoch X-ray images of models originating from the same initial density fields evolved in three different cosmologies: (1) an unbiased, low-density universe with OMEGA0 = 0.2; (2) an unbiased universe dominated by vacuum energy with OMEGA0 = 0.2 and lambda0 = 0.8 and (3) a biased Einstein-de Sitter model (OMEGA = 1, sigma8 = 0.59). Using measures of X-ray morphology such as the axial ratio and centroid shifting, we demonstrate that clusters evolved in the two low OMEGA0 models are much more regular, spherically symmetric, and centrally condensed than clusters evolved in the Einstein-de Sitter model. This morphology-cosmology connection, along with the availability of a large body of cluster X-ray observations, makes cluster X-ray morphology both a powerful and a practical cosmological discriminant.