Large Language Models (LLMs) are increasingly deployed in high-stakes domains such as clinical decision support and medical documentation. However, the robustness of these models against subtle linguistic variations, specifically stigmatizing language (SL) commonly found in human-authored clinical notes, remains critically under-explored. In this work, we investigate whether frontier LLMs inherit and propagate this human bias when processing clinical text. We systematically evaluate nine frontier LLMs across four stigmatized medical conditions, utilizing clinical vignettes injected with varying intensities and phenotypes of SL (doubt, blame, and maligning). Our results demonstrate that all evaluated models exhibit substantial bias, with clinical decision-making significantly skewed towards less aggressive patient management. Notably, we observe a high sensitivity to linguistic framing, where a single SL sentence is sufficient to alter model outputs, revealing a clear dose-response relationship. Furthermore, we evaluate standard prompt-based mitigation strategies, including Chain-of-Thought (CoT) reasoning and model self-debiasing. These approaches show limited efficacy; models struggle to explicitly identify SL while remaining implicitly influenced by it. Our findings expose a critical vulnerability in current LLMs regarding fairness and robustness in clinical NLP, underscoring the need for rigorous algorithmic guardrails to prevent the automation of health disparities.