Does Milnor K-Theory arise from Waldhausen K-Theory

I don't know if there any evidence for this to be true. Note that Quillen K-groups are defined as homotopy groups of some space (+-construction, Q-construction, Waldhausen construction etc), whereas Milnor K-groups were defined in terms of generators and relations,
which generalize generators and relations for classical K_2.

More invariantly Milnor K-groups can be constructed using homology of GL_n (paper of Suslin and Nesterenko) or as certain motivic cohomology groups of a field (Suslin-Voevodsky).
However, these constructions are unrelated to any homotopy groups.

Also, I'm not sure how you define Milnor K-theory for a general ring R?
(I was interpreting your question with "ring R" replaced by "field F".)