The scope of the Internet of Things (IoT) applications varies from strategic applications, such as smart grids, smart transportation, smart security, and smart healthcare, to industrial applications such as smart manufacturing, smart logistics, smart banking, and smart insurance. In the advancement of the IoT, connected devices become smart and intelligent with the help of sensors and actuators. However, issues and challenges need to be addressed regarding the data reliability and protection for significant next-generation IoT applications like smart healthcare. For these next-generation applications, there is a requirement for far-reaching privacy and security in the IoT. Recently, blockchain systems have emerged as a key technology that changes the way we exchange data. This emerging technology has revealed encouraging implementation scenarios, such as secured digital currencies. As a technical advancement, the blockchain network has the high possibility of transforming various industries, and the next-generation healthcare IoT (HIoT) can be one of those applications. There have been several studies on the integration of blockchain networks and IoT. However, blockchain-as-a-utility (BaaU) for privacy and security in HIoT systems requires a systematic framework. This paper reviews blockchain networks and proposes BaaU as one of the enablers. The proposed BaaU-based framework for trustworthiness in the next-generation HIoT systems is divided into two scenarios. The first scenario suggests that a healthcare service provider integrates IoT sensors such as body sensors to receive and transmit information to a blockchain network on the IoT devices. The second proposed scenario recommends implementing smart contracts, such as Ethereum, to automate and control the trusted devices' subscription in the HIoT services.