The abundance of complex and interconnected healthcare data offers numerous opportunities to improve prediction, diagnosis, and treatment. Graph-structured data, which includes entities and their relationships, is well-suited for capturing complex connections. Effectively utilizing this data often requires strong and efficient learning algorithms, especially when dealing with limited labeled data. It is increasingly important for downstream tasks in various domains to utilize self-supervised learning (SSL) as a paradigm for learning and optimizing effective representations from unlabeled data. In this paper, we thoroughly review SSL approaches specifically designed for graph-structured data in healthcare applications. We explore the challenges and opportunities associated with healthcare data and assess the effectiveness of SSL techniques in real-world healthcare applications. Our discussion encompasses various healthcare settings, such as disease prediction, medical image analysis, and drug discovery. We critically evaluate the performance of different SSL methods across these tasks, highlighting their strengths, limitations, and potential future research directions. Ultimately, this review aims to be a valuable resource for both researchers and practitioners looking to utilize SSL for graph-structured data in healthcare, paving the way for improved outcomes and insights in this critical field. To the best of our knowledge, this work represents the first comprehensive review of the literature on SSL applied to graph data in healthcare.