AI is having a significant effect in health through diagnostics, personalized medicine and surgeries. With great power comes responsibility, and there are simple questions of sustainability, ecological/environmental and ethical dimensions, even scaling out AI models. This power is accompanied by the obligations that surround providing AI innovations toward health care while minimizing ecological footprints and environmental effects and within the bounds of scalable sustainable development goals and eco-friendly offering(s). This paper is a systematic literature review of Green AI involvement in health toward a safe, sustainable and responsible proficiency for AI in health. While trade-offs in energy consumption exist, ethical issues exist around privacy, security, exploitation around algorithmic inequality/error when under powered were presented in this review of issuer. Sixty (60) papers of great interest, published between 2018 and early 2025 were reviewed. These papers demonstrated global trends of different research, but also demonstrated thematic concatenation across sustainability, privacy principles and ethical governance of AI in health systems. The findings included the emergence of technological advancements, e.g., federated learning, that secure patient data, while also encourage multi-disciplinary collaborations across clinician-devices-, policy- and ethical domains, and demonstrate the need for accountability. Finally, this paper enunciates the preparedness for future state of sustainable AI architectures for equitable health care innovations and the responsible governing of patient-initiated AI systems that are socially responsible and resilient within global health systems.