Yes. If you are given a simplicial set $X: \Delta^{\text{op}} \to \text{Sets}$, then
the the thick realization $||X||$  of $X$ is given by the same formula as the ordinary realization with the exception that one only uses injective order preserving maps of finite ordered sets. The map $||X|| \to |X|$ is always a homotopy equivalence (here's a reference: https://ncatlab.org/nlab/show/geometric+realization+of+simplicial+topological+spaces#GoodAndProper).

Now, given an ordered simplicial complex $K$,
the promotion $Ss(K)$ of $K$ to a simplicial set
has the property that $||Ss(K)|| = |K|$ identically, where $|K|$ means the geometric realization of $K$ as a simplicial complex. So,
$$
|K| = ||Ss(K)|| \simeq |Ss(K)|\, .
$$.