Large language models (LLMs) often follow conflicting reasoning paths on difficult tasks, creating uncertainty that is especially dangerous in clinical settings where errors can inform incorrect diagnoses or treatments. Standard self-consistency (SC) improves accuracy by voting over multiple reasoning chains, but it does not resolve the contradictions that produced the uncertainty. We propose Contrastive Adjudication via Selective Triggering (CAST), a lightweight inference-time method that detects disagreement across chain-of-thought samples and triggers contrastive adjudication only when model confidence falls below a threshold. Instead of re-asking the same question, CAST extracts the strongest argument for each competing reasoning path, presents them side by side, and asks the model to resolve the conflict definitively. Across multiple model families and three medical benchmarks, including HealthBench, a physician-rubric clinical task benchmark, CAST outperforms standard SC, compute-matched SC, SC-Max (with the maximum possible number of calls, usually 13), and the pooled CASC variant (which votes across all pre- and post- adjudication votes) across all datasets. Gains are largest on uncertain questions, reaching up to 29% over SC. These results, coupled with selective use of model calls, make CAST practical for clinical deployment where models must be accurate, efficient, and uncertainty-aware.