Lung cancer remains the leading cause of cancer-related death worldwide, and early diagnosis of lung cancer is critical for improving the survival rate of patients. Performing annual low-dose computed tomography (LDCT) screening among high-risk populations is the primary approach for early diagnosis. However, after each screening, whether to continue monitoring (with follow-up screenings) or to order a biopsy for diagnosis remains a challenging decision to make. Continuing with follow-up screenings may lead to delayed diagnosis but ordering a biopsy without sufficient evidence incurs unnecessary risk and cost. In this paper, we tackle the problem by an optimal stopping approach. Our proposed algorithm, called EarlyStop-RL, utilizes the structure of the Snell envelope for optimal stopping, and model-free deep reinforcement learning for making diagnosis decisions. Through evaluating our algorithm on a commonly used clinical trial dataset (the National Lung Screening Trial), we demonstrate that EarlyStop-RL has the potential to greatly enhance risk assessment and early diagnosis of lung cancer, surpassing the performance of two widely adopted clinical models, namely the Lung-RADS and the Brock model.