The model structure for complete Segal spaces is not right proper. To see this, one can first prove that the model structure for quasi-categories is not right proper: for instance, the map $\delta^1_2:\Delta_1\to\Delta_2$ is a fibration between fibrant objects in the Joyal model category (because it is the nerve of a fibration of the canonical model structure on the category of small categories), but its pullback along the inner horn $\Lambda^1_2\to\Delta_2$ is the boundary $\partial\Delta_1$. Now, given a quasi-category $X$, there is a canonical complete Segal space $N(X)$ associated to it (denoted like this because this is an homotopic version of the classical nerve): the space of $n$-simplices of $N(X)$ is the Kan complex $Map(\Delta_n,X)$, where $Map$ means the mapping space for the Joyal model category structure. A nice explicit model for $Map(\Delta_n,X)$ is just $k(\underline{Hom}(\Delta_n,X))$, where $\underline{Hom}$ is the internal Hom in simplicial sets, and where $k(A)$ denotes the maximal Kan complex contained in the quasi-category $A$. For this explicit model of $N(X)$, if $X=\Delta_m$, then we get that $N(X)$ is just the classical nerve of the poset corresponding to $\Delta_m$ (because there are no other isomorphisms than the identity in $\Delta_m$). In other words, $\Delta_m$ is a complete Segal space already. Therefore, the counter-example for right properness given above for quasi-categories gives a counter-example for complete Segal spaces. For the same reason, the model structure for Segal categories is not right proper.