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.  (**Edit:** in fact, it suffices to require the evident map to the pullback to be a weak equivalence only for $\delta:[0]\to[p]$ with $\delta(0)=p$.)

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.