From my understanding, one way to motivate Complete Segal Spaces is to see them as $(\infty,1)$-category objects inside the $(\infty,1)$-category of spaces, or rather $\infty$-groupoids, represented by Kan complexes. It looks like the definition of a CSS as a bisimplicial set satisfying some conditions does precisely this. However, if we follow the definition of a category object in an infinity-category at the nLab, we see a somewhat subtle difference.
Let $\mathbf{Kan}$ denote the category of Kan complexes, and $\infty \mathbf{Grpd}$ denote the $(\infty,1)$-category of $\infty$-groupoids, for instance represented by the simplicial localization of $\mathbf{Kan}$.
By definition, a CSS is a functor $\Delta^{op} \to \mathbf{Kan}$ satisfying Segal and completeness conditions, whereas an internal $(\infty,1)$-category to $\infty \mathbf{Grpd}$ is an $\infty$-functor $\Delta^{op} \to \infty \mathbf{Grpd}$ satisfying similar conditions.
The relevant nLab pages at Segal Space, Complete Segal Spaces and category object in an infinity-category seem to make no difference between the two notions. I do not doubt that the two definitions I have given here both present the theory of $(\infty,1)$-categories, but I couldn't find such an equivalence anywhere in the literature. To me it would seem like an interesting result in itself that any Complete Segal Object in the $(\infty,1)$-category $\infty \mathbf{Grpd}$ can be "strictified" to a Complete Segal Object in the category with weak equivalences $\mathbf{Kan}$.
I was wondering how to obtain such a strictification result, be it by an abstract argument or a direct construction.
As remarked by Dmitri P., if we define an $\infty$-category to be a relative category and an $\infty$-functor to be a weak equivalence-preserving functor, the question becomes tautological. Something similar seems to happen if we take $\mathbf{Kan}$-enriched categories as models : there the $\infty$-category of $\infty$-groupoids is the $\mathbf{Kan}$-enriched category of Kan complexes, and any $\mathbf{Kan}$-enriched functor has an underlying functor. In a sense, this is because relative categories and $\mathbf{Kan}$-enriched categories are already "strict enough models" of $\infty$-categories (they have strictly defined compositions, which are strictly unital and associative, etc.).
Hence I reformulate my question as follows : is there a similar strictification result if we work with quasi-categories or Complete Segal Spaces instead ? Can it be shown to follow from the case of relative categories or $\mathbf{Kan}$-enriched categories by general model category-theoretic arguments ?