As others have said, yes it's possible. Whether it's a good idea (esp. "in the spirit of cross platform development") is another matter.
Again, as has been noted, the language is not much without its libraries. Since Objective C is now used almost exclusively within a Mac/ iPhone context, being able to write code in it on Windows is not going to buy you much if they are your targets.
Furthermore, you can write Objective C on Windows already - I believe gcc, which is typically used with XCode, will compiler Obj C on Windows too, as well as GnuStep.
Which just leaves whether there is an advantage to running Objective C over .Net?
There are some nice features of Objective C that are not found in your typical .Net languages. Personally I really like the method name labelling of parameters.
However I don't think it's enough to make it worthwhile to bring it over. Most of the things Objective C is good at, C# has it's own solutions to. Now if only they would relent on refusing to implement named arguments...