Slightly puzzling: if you have an @me.com address that ought to be your login as that was how MobileMe worked. I'm not clear how you can have an Apple ID which is a Gmail address, but also an @me.com address. Possibly it may arise from the transition period between MobileMe and iCloud.
If you go to http://www.icloud.com , click on the cogwheel icon at bottom left, then in the pane which opens click on the 'Accounts' tab you will see something like my own accounts page:
Under 'Email addresses' when the top one in the list at left is selected there are three with the same text before the '@' (three because I started as DotMac; you evidently have two). These addresses cannot be deleted by you: I don't know whether Apple is able to delete them. You would have to ask as I indicated above.
There are three 'aliases' listed - these are the additional addresses which can be deleted, and which can exist in more than one version depending on when they were created: I'm assuming this isn't what you are talking about.