Closures
This is how I understand closure: assign a function to a variable and reuse it with different environments.
That's not a strictly accurate definition of a closure.
A closure is a function that has access to a referencing-environment. In Javascript, that means a function that is returned by another function and has access to the original functions scope. there are other SO questions that describe this very well
Closure's are general purpose structures that can be used in a variety of ways. One of their biggest benefits is that they protect private scope, which is why libraries like jQuery are often written as closures, so that they don't need to expose all their functions globally.
Promises
Promises are a different concept. They are a way of structuring asynchronous code to make it easier to follow the flow. A promise object in particular is an object that provides functions to chain operations in a clear and easy to read way. A promise might be implemented using closures, but it does not have to be. For instance here is an implementation that does not use closures:
https://gist.github.com/814052/690a6b41dc8445479676b347f1ed49f4fd0b1637
whereas jQuery's implementation uses at least one closure, but isn't really based on them
http://james.padolsey.com/jquery/#v=1.10.2&fn=jQuery.Deferred
Conclusion
Promises and Closures aren't directly related concepts. Closure's are a programming technique that might be used in a Promise implementation. In the end it is neither impossible or necessary to implement it like that.