Let us fix some notations. I will write $i(X)$ for the maximal Kan subcomplex of a quasi-category $X$ and $Hom(A,B)$ for the internal Hom of simplicial sets $A$ and $B$. If $B$ is a quasi-category, then so is $Hom(A,B)$ for any simplicial set $A$, and I will write $$i(A,B)=i(Hom(A,B))$$ It is a fact that (at least up to simplicial homotopy) any complete Segal space is of the form $i(\Delta^\ast,X)$ for a quasi-category $X$, where $\Delta^\ast$ is the standard cosimplicial object of simplicial sets (given by the Yoneda embedding); furthermore, the canonical inclusion $X\to i(\Delta^\ast,X)$ is a weal equivalence (where $X$ is considered as a (constant) simplicial quasi-category). Therefore, it is sufficient to understand morphisms of bisimplicial sets $N(C,W)\to i(\Delta^\ast,X)$ in the homotopy category of the Rezk model structure. The point is that these are homotopy classes of maps of bisimplicial sets (being maps from a cofibrant object to a fibrant one), and that those morphisms of bisimplicial sets can be understood rather explicitely in terms of morphisms of marked simplicial sets.
If $u:N(C)\to i(\Delta^\ast,X)$ is a morphism of bisimplicial sets, it is completely determined by the morphism of simplicial sets $u_0:N(C)\to X=i(\Delta^0,X)$X=i(\Delta^\ast,X)_0$. On the other hand, the map$u$sends any arrow of$W$to an invertible$1$-simplex of$X$if and only if the maps $$N(Hom([n],C))=Hom(\Delta^n,N(C))\to Hom(\Delta^n,X)$$ induced by$u_0$define maps $$N(C,W)_n\to i(\Delta^n,X)$$ which in turn define a morphism of bisimplicial sets $$N(C,W)\to i(\Delta^\ast,X)$$ Moreover, any map$N(C,W)\to i(\Delta^\ast,X)$is obtained in this way (I guess that interpreting all this in terms of marked simplicial sets would help to write down all this in a cleaner way). It is then easy to deduce from there that$N(C,W)$has the universal property of the localization of$N(C)$by$N(W)$in the homotopy category of the Rezk model structure: to deal with homotopies, you just have to replace$C$by$C\times I$, where$I$denotes the contractible groupoid with two objects (with an adequate subcategory of weak equivalences). 1 It seems to me that the answer is yes. Here is a sketchy argument. Let us fix some notations. I will write$i(X)$for the maximal Kan subcomplex of a quasi-category$X$and$Hom(A,B)$for the internal Hom of simplicial sets$A$and$B$. If$B$is a quasi-category, then so is$Hom(A,B)$for any simplicial set$A$, and I will write $$i(A,B)=i(Hom(A,B))$$ It is a fact that (at least up to simplicial homotopy) any complete Segal space is of the form$i(\Delta^\ast,X)$for a quasi-category$X$, where$\Delta^\ast$is the standard cosimplicial object of simplicial sets (given by the Yoneda embedding); furthermore, the canonical inclusion$X\to i(\Delta^\ast,X)$is a weal equivalence (where$X$is considered as a (constant) simplicial quasi-category). Therefore, it is sufficient to understand morphisms of bisimplicial sets$N(C,W)\to i(\Delta^\ast,X)$in the homotopy category of the Rezk model structure. The point is that these are homotopy classes of maps of bisimplicial sets (being maps from a cofibrant object to a fibrant one), and that those morphisms of bisimplicial sets can be understood rather explicitely in terms of morphisms of marked simplicial sets. If$u:N(C)\to i(\Delta^\ast,X)$is a morphism of bisimplicial sets, it is completely determined by the morphism of simplicial sets$u_0:N(C)\to X=i(\Delta^0,X)$. On the other hand, the map$u$sends any arrow of$W$to an invertible$1$-simplex of$X$if and only if the maps $$N(Hom([n],C))=Hom(\Delta^n,N(C))\to Hom(\Delta^n,X)$$ induced by$u_0$define maps $$N(C,W)_n\to i(\Delta^n,X)$$ which in turn define a morphism of bisimplicial sets $$N(C,W)\to i(\Delta^\ast,X)$$ Moreover, any map$N(C,W)\to i(\Delta^\ast,X)$is obtained in this way (I guess that interpreting all this in terms of marked simplicial sets would help to write down all this in a cleaner way). It is then easy to deduce from there that$N(C,W)$has the universal property of the localization of$N(C)$by$N(W)$in the homotopy category of the Rezk model structure: to deal with homotopies, you just have to replace$C$by$C\times I$, where$I\$ denotes the contractible groupoid with two objects (with an adequate subcategory of weak equivalences).