Vehicle crashes involve complex interactions between road users, split-second decisions, and challenging environmental conditions. Among these, two-vehicle crashes are the most prevalent, accounting for approximately 70% of roadway crashes and posing a significant challenge to traffic safety. Identifying Driver Hazardous Action (DHA) is essential for understanding crash causation, yet the reliability of DHA data in large-scale databases is limited by inconsistent and labor-intensive manual coding practices. Here, we present an innovative framework that leverages a fine-tuned large language model to automatically infer DHAs from textual crash narratives, thereby improving the validity and interpretability of DHA classifications. Using five years of two-vehicle crash data from MTCF, we fine-tuned the Llama 3.2 1B model on detailed crash narratives and benchmarked its performance against conventional machine learning classifiers, including Random Forest, XGBoost, CatBoost, and a neural network. The fine-tuned LLM achieved an overall accuracy of 80%, surpassing all baseline models and demonstrating pronounced improvements in scenarios with imbalanced data. To increase interpretability, we developed a probabilistic reasoning approach, analyzing model output shifts across original test sets and three targeted counterfactual scenarios: variations in driver distraction and age. Our analysis revealed that introducing distraction for one driver substantially increased the likelihood of "General Unsafe Driving"; distraction for both drivers maximized the probability of "Both Drivers Took Hazardous Actions"; and assigning a teen driver markedly elevated the probability of "Speed and Stopping Violations." Our framework and analytical methods provide a robust and interpretable solution for large-scale automated DHA detection, offering new opportunities for traffic safety analysis and intervention.