Arguably yes, but none of that is a good reason to put GNU in the name. I don't think even Stallman argued that Linux distributions should use the name GNU to give credit to GNU's influence.
The reason always given is a different one: it's because distros traditionally took a lot of code from the GNU project, which is a different matter. That reasoning does make some kind of sense, even though I don't fully agree.
POSIX specifies the API available to programs, and also shell and commands and stuff available to users. It does not specify which functions should be available from the standard C library, and which should be available from the kernel: from the standpoint of POSIX, it's all the same. POSIX doesn't care how the API is implemented, just that it is implemented correctly.