Molecular editing aims to modify a given molecule to optimize desired chemical properties while preserving structural similarity. However, current approaches typically rely on string-based or continuous representations, which fail to adequately capture the discrete, graph-structured nature of molecules, resulting in limited structural fidelity and poor controllability. In this paper, we propose MolEditRL, a molecular editing framework that explicitly integrates structural constraints with precise property optimization. Specifically, MolEditRL consists of two stages: (1) a discrete graph diffusion model pretrained to reconstruct target molecules conditioned on source structures and natural language instructions; (2) an editing-aware reinforcement learning fine-tuning stage that further enhances property alignment and structural preservation by explicitly optimizing editing decisions under graph constraints. For comprehensive evaluation, we construct MolEdit-Instruct, the largest and most property-rich molecular editing dataset, comprising 3 million diverse examples spanning single- and multi-property tasks across 10 chemical attributes. Experimental results demonstrate that MolEditRL significantly outperforms state-of-the-art methods in both property optimization accuracy and structural fidelity, achieving a 74\% improvement in editing success rate while using 98\% fewer parameters.