Clinical decision-making agents can benefit from reusing prior decision experience. However, many memory-augmented methods store experiences as independent records without explicit relational structure, which may introduce noisy retrieval, unreliable reuse, and in some cases even hurt performance compared to direct LLM inference. We propose GSEM (Graph-based Self-Evolving Memory), a clinical memory framework that organizes clinical experiences into a dual-layer memory graph, capturing both the decision structure within each experience and the relational dependencies across experiences, and supporting applicability-aware retrieval and online feedback-driven calibration of node quality and edge weights. Across MedR-Bench and MedAgentsBench with two LLM backbones, GSEM achieves the highest average accuracy among all baselines, reaching 70.90\% and 69.24\% with DeepSeek-V3.2 and Qwen3.5-35B, respectively. Code is available at https://github.com/xhan1022/gsem.