Let $f : X \to Y$ be a map of marked simplicial sets.
Then $f$ is a weak equivalence if and only if the induced functor on localizations of simplicial categories (where marked arrows are localized) is an equivalence.

Indeed, $f$ is a weak equivalence if and only if $R(f)$ is a weak equivalence, where $R$ is a fibrant replacement functor in the model category on marked simplicial sets. Since the forgetful functor from marked simplicial sets to simplicial sets with the Joyal model structure is the right part of a Quillen equivalence, $R(f)$ is a weak equivalence if and only if $R(f)$ induces a weak equivalence $U(R(f))$ of underlying quasicategories. Since $\mathfrak{C}$ is the left part of a Quillen equivalence, this map is a weak equivalence if and only if $\mathfrak{C}(U(R(F)))$ is a weak equivalence of simplicial categories. Thus, we just need to show that $\mathfrak{C}(U(R(X)))$ is the localization of $\mathfrak{C}(U(X))$ at the set of marked maps.

The underlying quasicategory of $R(X)$ can be defined as the localization of $X$ as described in section 1.1.3 of Dwyer-Kan localization revisited, Vladimir Hinich. The claim follows from the fact that functor $\mathfrak{C}$ maps the localization of a simplial set to DK localozation of the corresponding simplicial category, which is proved in the same paper right before Proposition 1.2.1.