Single-pixel imaging offers a means of overcoming the constraints imposed by the limited number of pixels in conventional imaging. The realization of phase imaging in a single-pixel framework with consistent stability and optimal accuracy remains challenging. In this study, we propose single-pixel phase imaging without interferometry and a phase iteration algorithm. The proposed methodology relies on extracting the zero-order intensity of the Fourier spectrum and experimentally achieves a repetition accuracy and error rate of 0.12% and 4.1%, respectively. These experimental findings indicate that our approach exhibits commendable stability and high accuracy and therefore holds great significance for single-pixel imaging development and classical phase imaging innovation.