We have investigated the spontaneous formation of the regular quadrilateral defect array in cholesteric emulsions, composed of water, surfactants, and cholesteric liquid crystals. Both at the substrate surface and at the surfactant-coated droplet surface, a homeotropic anchoring is enforced to the adjacent liquid crystal. We observe nucleation of numerous point defects, in the cholesteric planar texture immediately after termination of shear to the emulsion, which leads to the formation of the regular defect array. We examined the temporal changes in density, size, and the positional correlation of point defects through the formation process of the defect array by means of two-dimensional fast Fourier transformation (2D FFT) image analysis. Two distinct stages were identified in the formation process; the early stage is the nucleation and growth process of point defects, and the late stage is the rearrangement process of point defects to the regular defect array. Water droplets destabilize the arrangement of cholesteric helix, which induces the transition from the shear-induced planar alignment to the homeotropic orientation.