The C standard has this to say (5.1.2.3/5):
The least requirements on a conforming
implementation are:— At sequence points, volatile objects
are stable in the sense that previous
accesses are complete and subsequent
accesses have not yet occurred.— At program termination, all data
written into files shall be identical
to the result that execution of the
program according to the abstract
semantics would have produced.— The input and output dynamics of
interactive devices shall take place
as specified in
7.19.3.
So, without the standard library functions, the only behavior that a program is guaranteed to have, relates to the values of volatile objects, because you can’t use any of the guaranteed file access or “interactive devices”. “Pure C” only provides interaction via standard library functions.
Pure C isn’t the whole story, though, since your hardware could have certain addresses which do certain things when read or written (whether that be a SATA or PCI bus, raw video memory, a serial port, something to go beep, or a flashing LED). So, knowing something about your hardware, you can do a whole lot writing in C without using standard library functions. Potentially, you could implement the C standard library, although this might require access to special CPU instructions as well as special memory addresses.
But in pure C, with no extensions, and the standard library functions removed, you basically can’t do anything other than read the command line arguments, do some work, and return a status code from main. That’s not to be sniffed at, it’s still Turing complete subject to resource limits, although your only resource is automatic and static variables, no heap allocation. It’s not a very rich programming environment.
The standard libraries are part of the C language specification, but in any language there does tend to be a line drawn between the language “as such”, and the libraries. It’s a conceptual difference, but ultimately not a very important one in principle, because the standard says they come together. Anyone doing something non-standard could just as easily remove language features as libraries. Either way, the result is not a conforming implementation of C.
Note that a “freestanding” implementation of C only has to implement a subset of standard includes not including any of the I/O, so you’re in the position I described above, of relying on hardware-specific extensions to get anything interesting done. If you want to draw a distinction between the “core language” and “the libraries” based on the standard, then that might be a good place to draw the line.