<p>Is it possible for the .NET CLR to support Objective-C? Are there any reasons (either from a legal or implementational) standpoint why this wouldn't be possible?</p>
<p>In the spirit of cross-platform application development, it would be nice to be able to write and run Objective-C applications on Windows machines. At least I think it would.</p>

No, there really aren't.  .NET compiles to
IL
(or CIL or MSIL) which is basically machine code for a virtual machine.  You can program in IL like you do in assembly (not that I've seen anyone do it).  If you had the experience, knowledge, and gumption there would be no problem writing an Objective-C to IL compiler.  Hell, there are even
functional programming languages (F#)
for .NET.  Besides the messaging capabilities and the libraries, it's not too different from C#, VB.NET, etc.
What an awesome idea!  Objective-C# or some such.
Here's where you might start:
Wikipedia on Microsoft's IL