Building upon the strong sequence modeling capability, Generative Recommendation (GR) has gradually assumed a dominant position in the application of recommendation tasks (e.g., video and product recommendation). However, the application of Generative Recommendation in Point-of-Interest (POI) recommendation, where user preferences are significantly affected by spatiotemporal variations, remains a challenging open problem. In this paper, we propose Spacetime-GR, the first spacetime-aware generative model for large-scale online POI recommendation. It extends the strong sequence modeling ability of generative models by incorporating flexible spatiotemporal information encoding. Specifically, we first introduce a geographic-aware hierarchical POI indexing strategy to address the challenge of large vocabulary modeling. Subsequently, a novel spatiotemporal encoding module is introduced to seamlessly incorporate spatiotemporal context into user action sequences, thereby enhancing the model's sensitivity to spatiotemporal variations. Furthermore, we incorporate multimodal POI embeddings to enrich the semantic understanding of each POI. Finally, to facilitate practical deployment, we develop a set of post-training adaptation strategies after sufficient pre-training on action sequences. These strategies enable Spacetime-GR to generate outputs in multiple formats (i.e., embeddings, ranking scores and POI candidates) and support a wide range of downstream application scenarios (i.e., ranking and end-to-end recommendation). We evaluate the proposed model on both public benchmark datasets and large-scale industrial datasets, demonstrating its superior performance over existing methods in terms of POI recommendation accuracy and ranking quality. Furthermore, the model is the first generative model deployed in online POI recommendation services that scale to hundreds of millions of POIs and users.