Calcite is the most stable crystalline phase of calcium carbonate. It is applied or found in composite products, the food industry, biomineralization, archaeology, and geology, and its mechanical properties have attracted more and more attention. In this paper, the mechanical behaviors of single-crystal calcite under uniaxial tension in different directions were simulated with the molecular dynamics method. The obtained elastic moduli are in good agreement with the experimental results. It has been found from further research that single-crystal calcite has typical quasi-brittle failure characteristics, and its elastic modulus, fracture strength, and fracture strain are all strongly anisotropic. The tensile failure is caused by dislocation emission, void formation, and phase transition along the [010] and [421] directions, but by continuous dislocation glide and multiplication along the direction. The fracture strength, fracture strain, and elastic modulus are all sensitive to temperature, but only elastic modulus is not sensitive to strain rate. The effects of temperature and logarithmic strain rate on fracture strength are in good agreement with the predictions of fracture dynamics.