It's known that Cat with the Thomason model structure serves as a model for $\infty\mathrm{Grpd}$, and that RelCat has a corresponding model structure that serves as a model for $\infty\mathrm{Cat}$. (with Cat embedding in RelCat as those relative categories where everything is a weak equivalence)
In Barwick and Kan's paper, they define a homotopy relation generated by the natural transformations whose components are weak equivalences.
While appealing, this is inadequate; if I understand anything at all, the infinite zigzag category $Z$ depicted as
$$ \ldots \leftarrow \bullet \to \bullet \leftarrow \bullet \to \bullet \leftarrow \bullet \to \ldots $$
where all arrows are weak equivalences is supposed to have geometric realization homeomorphic to $\mathbb{R}$, and thus have the homotopy type of a point... but $Z$ is not homotopy equivalent to the terminal category, since any homotopy from $1_Z$ can only take a fixed, finite number of steps, but arbitrarily large steps are needed to connect every object to a specified one.
Every exposition on the topic I have seen simply punts the question over to simplicial sets or bisimplicial sets or similar: that whether or not a map is a weak equivalence is determined by the map it induces on nerves.
Is there a description of weak equivalences that can be phrased entirely within Cat or RelCat without taking a detour through simplicial sets or topological spaces?