DUSt3R-based end-to-end scene reconstruction has recently shown promising results in dense visual SLAM. However, most existing methods only use image pairs to estimate pointmaps, overlooking spatial memory and global consistency.To this end, we introduce GRS-SLAM3R, an end-to-end SLAM framework for dense scene reconstruction and pose estimation from RGB images without any prior knowledge of the scene or camera parameters. Unlike existing DUSt3R-based frameworks, which operate on all image pairs and predict per-pair point maps in local coordinate frames, our method supports sequentialized input and incrementally estimates metric-scale point clouds in the global coordinate. In order to improve consistent spatial correlation, we use a latent state for spatial memory and design a transformer-based gated update module to reset and update the spatial memory that continuously aggregates and tracks relevant 3D information across frames. Furthermore, we partition the scene into submaps, apply local alignment within each submap, and register all submaps into a common world frame using relative constraints, producing a globally consistent map. Experiments on various datasets show that our framework achieves superior reconstruction accuracy while maintaining real-time performance.