My guess, which has no experience to back it up, is that you probably could write Obj-C with Google's NDK somehow, given that GCC exists for ARM, is open source, has an Obj-C compiler and a basic Obj-C runtime (which if it doesn't already probably could be hacked up to work on a new architecture), etc.
That might also be a lot of work for questionable benefit.
And of course "Obj-C" (without the NS
classes) means something very different than "Cocoa", which is what most people really mean when they say "Obj-C". You might be able to re-use some of GNUstep for some that, but... Honestly, I doubt it. Sounds again like a lot of work.
So, yes, I think it is possible. It's also a lot of work and I don't think it's worth it.
Given what you've said, if I were attempting this, I would be tempted to write as much of your core logic as possible in C, then wrap it with two separate GUIs for each platform.