Abstract: We use a representative sample of 65 galaxy dusters observed with the Einstein IPC to constrain the range of cluster X-ray morphologies. We develop and apply quantitative and reproducible measures of cluster X-ray morphologies to constrain the intrinsic distributions of (1) emission-weighted centroid variation w(x), (2) emission-weighted axial ratio eta, (3) emission-weighted orientation theta(0), and (4) measures of the radial falloff, cr and beta. For each cluster we use a Monte Carlo procedure to determine the effects of Poisson noise, detector imperfections, and foreground/background X-ray point sources. We then use the range of cluster X-ray morphologies to constrain three generic cosmological models (Omega = 1, Omega(0) = 0.2, and Omega(0) = 0.2 and lambda(0) = 0.8). For each of these models, we evolve eight sets of Gaussian random initial conditions consistent with an effective power spectrum P(k) proportional to k(-1) on cluster scales. Using this sample of 24 numerical cluster simulations which include gravity and gas physics (but no cooling or ejection from galaxies), we compare the X-ray morphologies of the observed and simulated clusters. Specifically, we build artificial ensembles with the same distributions in the number of cluster photons, X-ray temperature, and cluster redshift as the Einstein ensemble; we then compare the observed and simulated distributions in w(x), eta, and alpha. The comparisons indicate that (1) these three morphological characteristics are sensitive to the underlying cosmological model and (2) galaxy clusters with the observed range of X-ray morphologies are very unlikely in low-Omega(0) cosmologies. The analysis favors the Omega = 1 model, though some discrepancies remain. We discuss the effects of changing the initial conditions and of including additional physics in the simulations.