I would say that it's possible, but this process must be crippled by many, many details.
ABI compatibility
The first thing to think of is Application Binary Interface compatibility. Unless you're able to call your functions the same way, the code is broken. So I guess (though I can't check at the moment) that compiling code with gcc on Linux/OS X and MinGW gcc on Windows should give the same binary code as far as no external functions are called. The problem here is that executable metadata may rely on some ABI assumptions.
Standard libraries
That seems to be the largest hurdle. Partly because of C preprocessor that can inline some procedures on some platforms, leaving them to run-time on others. Also, cross-platform dynamic interoperation with standard libraries is close to impossible, though theoretically one can imagine a code that uses a limited subset of the C standard library that is exposed through the same ABI on different platforms.
Static build mostly eliminates problems of interaction with other user-space code, but still there is a huge issue of interfacing with kernel: it's int $0x80
calls on x86 Linux and a platform-specifc set of syscall numbers that does not map to Windows in any direct way.
OS-specific register use
As far as I know, Windows uses register %fs for storing some OS-wide exception-handling stuff, so a binary compiled on Linux should avoid cluttering it. There might be other similar issues. Also, C++ exceptions on Windows are mostly done with OS exceptions.
Virtual addresses
Again, AFAIK Windows DLLs have some predefined address they're must be loaded into in virtual address space of a process, whereas Linux uses position-independent code for shared libraries. So there might be issues with overlapping areas of an executable and ported code, unless the ported position-dependent code is recompiled to be position-independent.
So, while theoretically possible, such transformation must be very fragile in real situations and it's impossible to re-plant the whole static build code - some parts may be transferred intact, but must be relinked to system-specific code interfacing with other kernel properly.
P.S. I think Wine is a good example of running binary code on a quite different system. It tricks a Windows program to think it's running in Windows environment and uses the same machine code - most of the time that works well (if a program does not use private system low-level routines or unavailable libraries).