Retrieval-augmented language models have exhibited promising performance across various areas of natural language processing (NLP), including fact-critical tasks. However, due to the black-box nature of advanced large language models (LLMs) and the non-retrieval-oriented supervision signal of specific tasks, the training of retrieval model faces significant challenges under the setting of black-box LLM. We propose an approach leveraging Fine-grained Feedback with Reinforcement Retrieval (FFRR) to enhance fact-checking on news claims by using black-box LLM. FFRR adopts a two-level strategy to gather fine-grained feedback from the LLM, which serves as a reward for optimizing the retrieval policy, by rating the retrieved documents based on the non-retrieval ground truth of the task. We evaluate our model on two public datasets for real-world news claim verification, and the results demonstrate that FFRR achieves significant improvements over strong LLM-enabled and non-LLM baselines.