LLM-as-a-Judge frameworks are increasingly trusted to automate evaluation in place of human experts, yet their reliability in high-stakes medical contexts remains unproven. We stress-test this assumption for detecting incomplete patient-facing medical responses, evaluating three rubric granularities (General-Likert, Analytical-Rubric, Dynamic-Checklist) and three backbone models across two clinician-annotated datasets, including HealthBench, the largest publicly available benchmark for medical response evaluation. LLM Judges discriminate complete from incomplete responses at and slightly above near chance (AUC $0.49$--$0.66$); at the threshold required to recall $90\%$ of incomplete responses, clinicians must still review the vast majority of the dataset, offering no triage utility. Even when model and clinician verdicts agree, they rarely cite the same explanation; and when they diverge, false positives stem from over-flagging non-essential gaps while false negatives reflect outright detection failures. These results reveal that LLM Judges and clinicians apply fundamentally different completeness standards; a finding that undermines their use as autonomous evaluators or triage filters in clinical settings.