We investigate the usefulness of generative Large Language Models (LLMs) in generating training data for cross-encoder re-rankers in a novel direction: generating synthetic documents instead of synthetic queries. We introduce a new dataset, ChatGPT-RetrievalQA, and compare the effectiveness of models fine-tuned on LLM-generated and human-generated data. Data generated with generative LLMs can be used to augment training data, especially in domains with smaller amounts of labeled data. We build ChatGPT-RetrievalQA based on an existing dataset, human ChatGPT Comparison Corpus (HC3), consisting of public question collections with human responses and answers from ChatGPT. We fine-tune a range of cross-encoder re-rankers on either human-generated or ChatGPT-generated data. Our evaluation on MS MARCO DEV, TREC DL'19, and TREC DL'20 demonstrates that cross-encoder re-ranking models trained on ChatGPT responses are statistically significantly more effective zero-shot re-rankers than those trained on human responses. In a supervised setting, the human-trained re-rankers outperform the LLM-trained re-rankers. Our novel findings suggest that generative LLMs have high potential in generating training data for neural retrieval models. Further work is needed to determine the effect of factually wrong information in the generated responses and test our findings' generalizability with open-source LLMs. We release our data, code, and cross-encoders checkpoints for future work.