Large language models in healthcare often produce emotionally flat or opaque responses, failing to provide the transparent reasoning required for clinical trust. We present RECAP (Reflect-Extract-Calibrate-Align-Produce), an inference-time framework grounded in cognitive appraisal theory that decomposes patient input into auditable, appraisal-theoretic stages without retraining. Across multiple benchmarks and models from 8B to 120B parameters, RECAP improves alignment with human judgments, with gains inversely proportional to model scale. Intermediate outputs further reveal that models systematically underweight relational factors such as social support. In blinded evaluations, oncology fellows rated RECAP responses significantly higher than baselines with 76-88% win rates, demonstrating that principled prompting can enhance medical AI's emotional intelligence while maintaining the transparency required for clinical deployment.