Predictive process monitoring (PPM) aims to predict the future behavior of process instances to mitigate process violations or take preventive measures. Current PPM methods for next event prediction (NEP) often utilize machine learning techniques, while first approaches also use deep learning techniques, especially natural language processing (NLP). Hence, these approaches often require extensive data preprocessing. To counteract this, we train and evaluate a fine-tuned large language model (LLM) to directly generate NEPs from XES-formatted event logs without any preprocessing. The results suggest that the proposed PPM approach performs comparably to the state-of-the-art in ML-based PPM, while contributing a simplified prediction process for NEP with minimal data preprocessing. Additionally, our LLM-driven approach produces valid XES outputs in nearly all cases, facilitating the direct export of predictions as event logs to be processed downstream (e.g., to employ process mining techniques or simulation). Further, our method offers easy integration into existing organizational infrastructures.