COVID-19 has emerged as a highly contagious disease which has caused a devastating impact across the world with a very large number of infections and deaths. Timely and accurate testing is paramount to an effective response to this pandemic as it helps identify infections and therefore mitigate (isolate/cure) them. In this paper, we investigate this challenge and contribute by presenting a blockchain-based solution that incorporates self-sovereign identity, re-encryption proxies, and decentralized storage, such as the interplanetary file systems (IPFS). Our solution implements digital medical passports (DMP) and immunity certificates for COVID-19 test-takers. We present smart contracts based on the Ethereum blockchain written and tested successfully to maintain a digital medical identity for test-takers that help in a prompt trusted response directly by the relevant medical authorities. We reduce the response time of the medical facilities, alleviate the spread of false information by using immutable trusted blockchain, and curb the spread of the disease through DMP. We present a detailed description of the system design, development, and evaluation (cost and security analysis) for the proposed solution. Since our code leverages the use of the on-chain events, the cost of our design is almost negligible. We have made our smart contract codes publicly available on Github.