The traditional Pound-Drever-Hall (PDH) technique utilizes analog devices to actively stabilize the frequency of lasers. However, this results in a bulky system and a rigid control process, making it difficult to meet the requirements of miniaturization and automation of the frequency stabilization system for new applications such as space gravitational wave detection. In this paper, an automatic peak-finding algorithm based on backward difference is specifically designed for frequency discrimination signal peak search, which effectively reduces human intervention in the frequency stabilization process. This method identifies the main signal peak and controls state switching by comparing the time width of consecutive signal peaks. Moreover, it avoids the inherent drawbacks of the conventional thresholding method. We have also designed and built a digital frequency stabilization system based on a field-programmable gate array (FPGA). This system digitizes and integrates the discrete components of the stabilization servo feedback control into a single FPGA, forming a fast servo feedback loop with a piezoelectric actuator. The digital frequency stabilization system first obtains the frequency discrimination signal locally through an amplitude demodulation, and then achieves automatic peak-finding through the designed backward difference algorithm. Finally, the servo controller is activated at the lock-in point, and an incremental digital PID algorithm is used to successfully lock the frequency of a commercial Nd:YAG laser to a resonance of a 10 cm Fabry-Pérot cavity with a finesse of 350000. During functional testing, the system maintained frequency lock for half an hour, with a wavelength meter measurement showing a relative frequency drift of less than 2 MHz. This result validates the effectiveness of the designed automatic peak-finding algorithm and underscores the merits of FPGA as an effective approach for achieving comprehensive digital laser frequency stabilization control.