Like Rust, C defines what happens if you don't have its rich standard runtime, you get "free standing" C.
C's free standing mode is even thinner than Rust's core, because it doesn't supply a library of code, you just get the primitive types, the operators and the language features that don't involve any libraries. To give a very concrete example, Rust's core depends on memcmp() existing, Rust assumes your toolchain knows how to memcmp() on your target architecture inherently, but in C you could write memcmp() if you had to. You won't these days, because your C compiler invariably provides this feature, but in principle you could.
In principle C++ also defines a "free standing" mode, but it's a mess and so people don't write for C++ free standing mode in the real world. C++ in an environment where you can't have the actual standard library is likely to be specified in terms of which features from the standard library you can have and which you cannot, for each such environment. For example maybe you can have threading but no filesystem APIs, or you can have a heap allocator but no threads.