Tensile strength is an important parameter that affects the initiation and propagation of shale reservoir fractures during hydraulic fracturing. Shale is often filled with minerals such as calcite. To explore the effect of calcite minerals on the tensile strength and failure mode of shale, in this paper, lower Cambrian shale cores were observed by microslice observations and core X-ray whole-rock mineral diffraction analysis, and 7 groups of numerical direct tensile tests were performed on simulated shale samples with different azimuth angles. The test results show that as the azimuth angle α increases, the tensile strength of the samples gradually decreases, and the fracture rate also shows a decreasing trend. The failure modes can be summarized as root-shaped (0° and 15°), step-shaped (30 and 45°), fishbone-shaped (60°), and river-shaped (75° and 90°) fracturing. The smaller the azimuth angle α, the easier it is for hydraulic fractures to propagate along the direction of the calcite veins and inhibit the formation of fracture networks in the shale matrix. Considering the correlation between the acoustic emission characteristics and failure mode, the fractal dimension is used to reflect the microscopic failure mode of shale. The larger the fractal dimension, the higher the fracture rate is, the more microcracks exist at the edge of the main crack, the more severe the internal damage is, and the more complex the failure mode of the sample is. As the azimuth angle α increases, the fractal dimension shows a decreasing trend, and the crack becomes smoother. This research has important reference value for the study of hydraulic fracture initiation mechanisms and natural fracture propagation.