Let $i : \mathbf{Grp}\to \mathbf{Mon}$ denote the forgetful functor from groups to monoids. It has a left adjoint, $(-)^{gp}$, which one could call group completion.
We have an induced functor $\mathbf{sMon}\to \mathbf{sGrp}$ between simplicial monoids and simplicial groups, which is again left adjoint to the inclusion. I'll use the same notation for the two functors.
Does $(-)^{gp} : \mathbf{sMon\to sGrp}$ preserve weak equivalences ?
Motivation : with the usual model structures on both sides, the forgetful functor $\mathbf{sGrp\to sMon}$ is clearly right Quillen, and preserves weak equivalences, so its right-derived functor can be "computed as itself".
On the other hand, I'd like to know if the same can be said of the left-derived functor of $(-)^{gp}$. The homotopical version of the group completion of a topological monoid $M$ is usually defined as $\Omega BM$; and it is (at the derived level) a left adjoint to the inclusion of grouplike $E_1$-spaces into $E_1$-spaces - I'm interested in knowing how far apart those constructions are.
I think I can already prove a partial result which encompasses the abelian case and also the connected case : if $f : M\to N$ is a weak equivalence and $H_*(M), H_*(N)$ satisfy the Ore condition with respect to the image of $\pi_0(M)$ (resp. $\pi_0(N)$) under $\pi_*(|M|)\to H_*(M)$ then $f^{gp} : M^{gp}\to N^{gp}$ is a weak equivalence (this is because $|M^{gp}|, |N^{gp}|$ are simple spaces, so it suffices to prove that $f^{gp}$ is a homology equivalence, and the Ore condition allows one to give an easy description of $H_*(M^{gp})$)
No, not at all.
For a counterexample, take the nerve of the skeleton of the symmetric monoidal category of finite sets and their bijections.
This is a simplicial monoid, whose homotopy group completion is weakly equivalent to the sphere spectrum by the Barratt–Priddy–Quillen theorem. The sphere spectrum has nontrivial higher homotopy groups, whereas the (strict) group completion does not.
For a positive answer, one may observe that weak equivalences between retracts of free simplicial monoids are preserved.