The increased number of Internet services and the exposure of the personal information to the public network, create a problem with the privacy of the personal data. This problem is extremely impactful in the healthcare systems where the sensitive medical data circulate in the network. The emerging Blockchain technologies offer properties and tools for better personal data access management. The integration of the Blockchain technologies into the healthcare systems can provide better data privacy. However, beside the attractive properties, the current Blockchain technologies cannot achieve sufficient scalability to serve a global public healthcare system and cannot provide compatibility with the GDPR and HIPAA regulations. To overcome these limitations, this paper introduces new deducted Blockchain-based public healthcare system, the BloHeS, with main focus on several functional entities, such as optimal consensus mechanism, better storage organization, improved network organization and compatibility with GDPR and HIPAA regulations. It also defines a procedure for patient-centric data access management, protocols for Island management and new block structure.