Quillens higher Kgroups of rings can be realized as π_{n}K(C)  the Waldhausen KTheory of a suitable Waldhausen category C. Is this also true for Milnor KTheory of Rings? Is there a functor F from rings to waldhausen categories s.t. $K^M_n(R)\cong \pi_n(K(F(R))$?

I don't know if there any evidence for this to be true. Note that Quillen Kgroups are defined as homotopy groups of some space (+construction, Qconstruction, Waldhausen construction etc), whereas Milnor Kgroups were defined in terms of generators and relations, which generalize generators and relations for classical K_2. More invariantly Milnor Kgroups can be constructed using homology of GL_n (paper of Suslin and Nesterenko) or as certain motivic cohomology groups of a field (SuslinVoevodsky). However, these constructions are unrelated to any homotopy groups. Also, I'm not sure how you define Milnor Ktheory for a general ring R? (I was interpreting your question with "ring R" replaced by "field F".) 


Bob Thomason proved that there is no Milnor Ktheory functor for schemes, with a reasonable map to Quillen Ktheory, in:



The original question seems not to have been answered yet. One answer might be that it would be unnatural to expect all the Milnor Kgroups of a field R to arise as the homotopy groups of a single space $K(F(R))$, because the natural way they currently arise is as homotopy groups of separate spaces, or better, of separate spectra. The spectra are the EilenbergMacLane spectra $\mathbb Z(n)$ associated to the chain complexes that compute motivic cohomology of $R$, namely, $K_n^M R = \pi_{n} \mathbb Z(n)$. 

