The cloud computing technology has emerged, developed, and matured in recent years, consequently commercializing remote outsourcing storage services. An increasing number of companies and individuals have chosen the cloud to store their data. However, accidents, such as cloud server downtime, cloud data loss, and accidental deletion, are serious issues for some applications that need to run around the clock. For some mission and business-critical applications, the continuous availability of outsourcing storage services is also necessary to protect users' outsourced data during downtime. Nevertheless, ensuring the continuous availability of data in public cloud data integrity auditing protocols leads to data privacy issues because auditors can obtain the data content of users by a sufficient number of storage proofs. Therefore, protecting data privacy is a burning issue. In addition, existing data integrity auditing schemes that rely on semi-trusted third-party auditors have several security problems, including single points of failure and performance bottlenecks. To deal with these issues, we propose herein a blockchain-based continuous data integrity checking protocol with zero-knowledge privacy protection. We realize a concrete construction by using a verifiable delay function with high efficiency and proof of retrievability, and prove the security of the proposal in a random oracle model. The proposed construction supports dynamic updates for the outsourced data. We also design smart contracts to ensure fairness among the parties involved. Finally, we implement the protocols, and the experimental results demonstrate the efficiency of the proposed protocol.