Vesicles enriched in certain negatively charged lipids, such as phosphatidylserine and PIP2, are known to undergo fusion in the presence of calcium ions without assistance from protein assemblies. Other lipids do not exhibit this propensity, even if they are negatively charged. Using our recently developed methodology, we extract elastic properties of a representative set of lipids. This allows us to trace the origin of lipid-calcium selectivity in membrane fusion to the formation of lipid clusters with long-range correlations that induce negative curvature on the membrane surface. Furthermore, the clusters generate lateral tension in the headgroup region at the membrane surface, concomitantly also stabilizing negative Gaussian curvature. Finally, calcium binding also reduces the orientational polarization of water around the membrane head groups, potentially reducing the hydration force acting between membranes. Binding calcium only weakly increases membrane bending rigidity and tilt moduli, in agreement with experiments. We show how the combined effects of calcium binding to membranes lower the barriers along the fusion pathway that lead to the formation of the fusion stalk as well as the fusion pore.