Probably worth mentioning that Miguel de Icaza (CTO of Xamarin) at least considered that Xamarin support a single distro of Linux in 2011. He's pretty approachable, and often answers questions on IRC. It might be neat to ask if they've revisited that option, even if it didn't support any sort of GUI RAD.
Just as an alternative to, "No, you absolutely can't develop for Xamarin on Linux," I've often considered using MonoDevelop on Linux to develop my faceless, shareable code. In a way, I already do something similar when I'm using my Windows laptop. You can't use Xamarin Studio to develop for iOS on Windows, so I stub out a plain console project and have it call my controllers that live in another library project. I prefer Visual Studio to Xamarin Studio (and am using Starter & Indie licenses, so I can't use Xamarin for Visual Studio), so I like to do faceless development on Windows. Then, when I'm "done" with (ie, ready to take a break from) the faceless dev, I head over to Xamarin Studio on OS X to hook things up to the iOS UI stubs I made there.
It'd be easy enough to do that on Linux "for Android", and only use Windows or OS X when you develop your UI and create your distributable. Admittedly, the UI is usually a ton of code (always more than I expect, at least), but you could easily get away with good, well-factored libs developed on Linux for let's say half of your coding or more. And it forces you to separate your concerns nicely, which I always appreciate.
EDIT: I feel someone should point out that Mono-qua-Mono does run on Linux already. It's just the Xamarin stuff -- the glue to the more popular consumer platforms -- that isn't on Linux. So you can run Xamarin Mono on Linux just fine right now, and have been able to longer than Xamarin's been "a thing". ;^)