In Mac OS X application software development, do you have to use Objective-C when compiling and linking an executable format?
You can use C or C++ too to compile binaries and executables.  Objective-C is mainly the source for developing for the MacOS user interface (Cocoa), but even there is still the old school interface (Carbon) available, just for the latter sample projects and docs are hard to find these days.

But if you want to create a tool that works like classic unix tools (stdin/stdout) or a shared library, C or C++ are absolutely viable solutions. E.g. you can download Xcode and from the Gui or command line (gcc) you can easily compile a simple int main(int, char**) { printf("Hello World!"); return 0; } program.

Also, Objective-C is just a C-extension, so you don't have to use its features.