Large language models (LLMs) have shown remarkable capabilities in various natural language processing tasks, yet they often struggle with maintaining factual accuracy, particularly in knowledge-intensive domains like healthcare. This study introduces LEAF: Learning and Evaluation Augmented by Fact-Checking, a novel approach designed to enhance the factual reliability of LLMs, with a focus on medical question answering (QA). LEAF utilizes a dual strategy to enhance the factual accuracy of responses from models such as Llama 3 70B Instruct and Llama 3 8B Instruct. The first strategy, Fact-Check-Then-RAG, improves Retrieval-Augmented Generation (RAG) by incorporating fact-checking results to guide the retrieval process without updating model parameters. The second strategy, Learning from Fact-Checks via Self-Training, involves supervised fine-tuning (SFT) on fact-checked responses or applying Simple Preference Optimization (SimPO) with fact-checking as a ranking mechanism, both updating LLM parameters from supervision. These findings suggest that integrating fact-checked responses whether through RAG enhancement or self-training enhances the reliability and factual correctness of LLM outputs, offering a promising solution for applications where information accuracy is crucial.