Automated approaches to answer patient-posed health questions are rising, but selecting among systems requires reliable evaluation. The current gold standard for evaluating the free-text artificial intelligence (AI) responses--human expert review--is labor-intensive and slow, limiting scalability. Automated metrics are promising yet variably aligned with human judgments and often context-dependent. To address the feasibility of automating the evaluation of AI responses to hospitalization-related questions posed by patients, we conducted a large systematic study of evaluation approaches. Across 100 patient cases, we collected responses from 28 AI systems (2800 total) and assessed them along three dimensions: whether a system response (1) answers the question, (2) appropriately uses clinical note evidence, and (3) uses general medical knowledge. Using clinician-authored reference answers to anchor metrics, automated rankings closely matched human ratings. Our findings suggest that carefully designed automated evaluation can scale comparative assessment of AI systems and support patient-clinician communication.