The integration of voice-based AI agents in healthcare presents a transformative opportunity to bridge economic and accessibility gaps in digital health delivery. This paper explores the role of large language model (LLM)-powered voice assistants in enhancing preventive care and continuous patient monitoring, particularly in underserved populations. Drawing insights from the development and pilot study of Agent PULSE (Patient Understanding and Liaison Support Engine) -- a collaborative initiative between IBM Research, Cleveland Clinic Foundation, and Morehouse School of Medicine -- we present an economic model demonstrating how AI agents can provide cost-effective healthcare services where human intervention is economically unfeasible. Our pilot study with 33 inflammatory bowel disease patients revealed that 70\% expressed acceptance of AI-driven monitoring, with 37\% preferring it over traditional modalities. Technical challenges, including real-time conversational AI processing, integration with healthcare systems, and privacy compliance, are analyzed alongside policy considerations surrounding regulation, bias mitigation, and patient autonomy. Our findings suggest that AI-driven voice agents not only enhance healthcare scalability and efficiency but also improve patient engagement and accessibility. For healthcare executives, our cost-utility analysis demonstrates huge potential savings for routine monitoring tasks, while technologists can leverage our framework to prioritize improvements yielding the highest patient impact. By addressing current limitations and aligning AI development with ethical and regulatory frameworks, voice-based AI agents can serve as a critical entry point for equitable, sustainable digital healthcare solutions.