I am learning about blockchain and I've read that blockchains are immutable. Suppose my health information is stored on a block and I have gotten diagnosed with a disease and now my information needs to be updated. I have read online (here) that to do this we have to create a new block with the updated data and the previous block is marked invalid. But now isn't it possible that someone could add a new block with my incorrect data and claim that it is the updated one and then my original data block would be invalid?
Yes, absolutely. You are right. You can think of a block-chain like an "append-only log": anyone can append a log entry saying anything. That will give you a reasonable intuition.
If you want to use it for something useful, you'll probably need to introduce extra checks. For instance, perhaps you can sign your health record with your private key. To update it, you append a new block that has new information, also signed. If we agree to only use the most recent (last) block and ignore all previous ones, this lets you update prior information by appending a new block. Also, because of the digital signature, someone else who doesn't know your private key then can't append something with the correct signature. Exactly what extra stuff you need will depend on the particular application.
Blockchain is not magic. Contrary to what some might have you believe, it doesn't somehow solve all the security problems in these various applications. It is just a technical tool that can be used in combination with many other techniques as part of a larger system.