Yes. Let $W$ be a complete Segal space, thought of as a simplicial "space" $(W_q)$. The fibrant objects of your model category will be the fibrations $f:X\to W$ such that for each simplicial operator $\delta:[q]\to [p]$ with $\delta(q)=p$, the evident map from $X_p$ to the pullback of $$X_q \xrightarrow{f} W_q \xleftarrow{\delta} W_p$$ is a weak equivalence of spaces.
I worked out some of this years ago, but never finished it; somebody should do this (or perhaps someone has already?). Lurie has done pretty much exactly the same thing in the context of quasi-categories, in HTT.