The pandemic crisis and the resulting global uncertainties have obviously had a severe impact on the healthcare supply chain (HSC), leading scholars, healthcare executives, and policymakers to focus on the sustainability of the HSC. Technologies have emerged and developed rapidly in recent years, especially in the healthcare industry, for coping with the pandemic crisis and supporting the “new normal” for humankind. Within this context, various new technologies have been implemented to maximize the supply chain process, ensure patient and healthcare worker safety, and improve the quality of care. Hence, the integration of a technological dimension with the traditional three pillars of sustainability may aid in attempts to define the potential attributes of these dimensions of sustainability. Therefore, this study aimed to identify the key attributes of a sustainable healthcare supply chain (SHSC), and this paper presents a new, four-dimensional model for SHSCs, consisting of social, environmental, economic, and technological dimensions. A systematic literature review was conducted, resulting in the identification of 35 potential SHSC attributes. The Fuzzy Delphi Method (FDM) was then applied to determine the appropriateness of these potential attributes according to the consensus of 13 experts, including healthcare workers in a variety of medical specialties, who profoundly understand HSC sustainability. The results yielded 22 appropriate attributes, which were then categorized across the four dimensions. Consequently, a new model of an SHSC, which prioritizes patient safety, was constructed and is proposed here. This SHSC model can be applied strategically to the healthcare industry to enhance the safety of both medical personnel and patients in a sustainable manner.