This paper explores the significant impact of AI-based medical devices, including wearables, telemedicine, large language models, and digital twins, on clinical decision support systems. It emphasizes the importance of producing outcomes that are not only accurate but also interpretable and understandable to clinicians, addressing the risk that lack of interpretability poses in terms of mistrust and reluctance to adopt these technologies in healthcare. The paper reviews interpretable AI processes, methods, applications, and the challenges of implementation in healthcare, focusing on quality control to facilitate responsible communication between AI systems and clinicians. It breaks down the interpretability process into data pre-processing, model selection, and post-processing, aiming to foster a comprehensive understanding of the crucial role of a robust interpretability approach in healthcare and to guide future research in this area. with insights for creating responsible clinician-AI tools for healthcare, as well as to offer a deeper understanding of the challenges they might face. Our research questions, eligibility criteria and primary goals were identified using Preferred Reporting Items for Systematic reviews and Meta-Analyses guideline and PICO method; PubMed, Scopus and Web of Science databases were systematically searched using sensitive and specific search strings. In the end, 52 publications were selected for data extraction which included 8 existing reviews and 44 related experimental studies. The paper offers general concepts of interpretable AI in healthcare and discuss three-levels interpretability process. Additionally, it provides a comprehensive discussion of evaluating robust interpretability AI in healthcare. Moreover, this survey introduces a step-by-step roadmap for implementing responsible AI in healthcare.