Large language models (LLMs) increasingly serve as interactive healthcare resources, yet user acceptance remains underexplored. This study examines how ease of use, perceived usefulness, trust, and risk perception interact to shape intentions to adopt DeepSeek, an emerging LLM-based platform, for healthcare purposes. A cross-sectional survey of 556 participants from India, the United Kingdom, and the United States was conducted to measure perceptions and usage patterns. Structural equation modeling assessed both direct and indirect effects, including potential quadratic relationships. Results revealed that trust plays a pivotal mediating role: ease of use exerts a significant indirect effect on usage intentions through trust, while perceived usefulness contributes to both trust development and direct adoption. By contrast, risk perception negatively affects usage intent, emphasizing the importance of robust data governance and transparency. Notably, significant non-linear paths were observed for ease of use and risk, indicating threshold or plateau effects. The measurement model demonstrated strong reliability and validity, supported by high composite reliabilities, average variance extracted, and discriminant validity measures. These findings extend technology acceptance and health informatics research by illuminating the multifaceted nature of user adoption in sensitive domains. Stakeholders should invest in trust-building strategies, user-centric design, and risk mitigation measures to encourage sustained and safe uptake of LLMs in healthcare. Future work can employ longitudinal designs or examine culture-specific variables to further clarify how user perceptions evolve over time and across different regulatory environments. Such insights are critical for harnessing AI to enhance outcomes.