It sounds like you're using IMAP push, and in that case, the app is trying to start up its push process. And yes, Android -- if done by the book -- strongly recommends putting an icon up (since 2.0). There is a setting to turn it off, though.
I still don't know why it would do that if there really is no connection (like I said, the code does have a check for this)... Did you change the setting in the app to not check for a connection before doing its thing?
@jacasares: those are very wise words, but what if I'm running into a networking bug on 4.1-4.2, which manifests differently on two official Google devices (GNex/Nexus7), connected to the same WiFI router, logged into the same mail server? That's a rhetorical question, of course