The aging of the world’s population and the growth in the number of people with chronic diseases have increased expenses with medical care. Thus, the use of technological solutions has been widely adopted in the medical field to improve the patients’ health. In this context, approaches based on Cloud Computing have been used to store and process the information generated in these solutions. However, using Cloud can create delays that are intolerable for medical applications. Thus, the Fog Computing paradigm emerged as an alternative to overcome this problem, bringing computation and storage closer to the data sources. However, managing medical data stored in Fog is still a challenge. Moreover, characteristics of availability, performance, interoperability, and privacy need to be considered in approaches that aim to explore this problem. So, this article shows a software architecture based on Fog Computing and designed to facilitate the management of medical records. This architecture uses Blockchain concepts to provide the necessary privacy features and to allow Fog Nodes to carry out the authorization process in a distributed way. Finally, this paper describes a case study that evaluates the performance, privacy, and interoperability requirements of the proposed architecture in a home-centered healthcare scenario.