This request is mostly covered here, but the answer is basically "this is not supported right now" and I did not see a feature request for it: https://answers.atlassian.com/questions/262527/bamboo-deployments-approved-and-broken

While the release review feature is nice, it could provide more functionality. Right now it looks like anyone who has permissions to view the deployment project can say a release is approved or broken. I would like the ability to restrict this to, for example, our QA department and release manager.

Additionally, it would be really nice if you could restrict the deployment process to only allow approved releases to be promoted to another environment. For example, say I have a new release to our Development environment that was deployed automatically from our build plan. I want the ability to have our QA team or release manager approve that build, and then, only once it has been deployed to Development and approved could it be deployed to the QA environment. QA checks it out, approves it, and only then can it be moved into UAT, Staging, Production, etc.