A paper by Mcduff and Segal justifies the following definition: A map of h-spaces $X \to Y$ is a group completion if the map is a localization on homology.

In the paper they prove that when $X$ is a topological monoid, the canonical map $X \to \Omega B X$ is a group completion in the above sense.

Unfortunately the example where $X=\sqcup_n B\Sigma_n $ shows that neither the space $Y$, nor the map, is unique up to homotopy:

Here the map $X \to Y=\mathbb{Z} \times B \Sigma_\infty$ is a group completion. So is the map $X \to Y_+=\mathbb{Z} \times (B \Sigma_\infty)_+$. $Y$ has homotopy groups that are concentrated in degree 1, whereas $Y_+$ has as its homotopy groups the stable homotopy groups of spheres.

In this case the map $X \to Y \to Y_+$ and $X \to Y_+$ are the same up to homotopy. This raises the question

Question: Given an H-space $X$, is any group completion $X \to Y$ followed by the plus construction map going to be unique up to homotopy?