rustc can use a few different backends. By my understanding, the LLVM backend is fully supported, the Cranelift backend is either fully supported or nearly so, and there's a GCC backend in the works. In addition, there's a separate project to create an independent Rust frontend as part of GCC.

Even then, there are still some systems that will support C but won't support Rust any time soon. Systems with old compilers/compiler forks, systems with unusual data types which violate Rust's assumptions (like 8 bit bytes IIRC)