Is the D language completely dependant upon the D runtime?

Yes, indeed, you can implement the functions of DRuntime that the compiler expects right in your main module (or wherever), compile without a runtime, and it'll Just Work (tm).

If you just build your code without a runtime, the compiler will emit errors when it's missing a symbol that it expects to be implemented by the runtime. You can then go and look at how DRuntime implements it to see what it does, and then implement it in whatever way you prefer. This is what XOmB, the kernel written in D (language version 1, though, but same deal), does: http://xomb.net/index.php?title=Main_Page

A lot of DRuntime isn't actually used by many applications, but it's the most convenient way to include the runtime components of D into applications, so that's why it's done as a static library (hopefully a shared library in the future).

It's pretty much the same as C and C++ I expect. The language itsself compiles to native code and just runs. But there is some code that is always needed to set everything up to run your program, for example processing command line parameters.

And some more complex language facilities are better implemented by calling some standard code rather than generating the code everywhere it is used. For example throwing an exception needs to find the relevent handler function. No doubt the compiler could insert the code to do there everywhere it was used, but it's much more sensible to write the code in a library and call that. Plus there are many pre-written library functions in the standard library.

All of this taken together is the runtime.

If you write C you can use it to write an operating system because you can write the startup code yourself, you can write all the code for handing memory allocation yourself, you can write all the code for standard functions like strcat yourself instead of using the provided ones in the runtime. But you'd not want to do that for any application program.