None of the trail is paved, and very narrow road bike tires are not ideal for the rough surface. However, the surface is generally the better the farther west you are on the trail, and some have ridden road bikes on it (e.g. between Cedar Falls and Snoqualmie Pass). Patches of deeper gravel in sections between Easton and Ellensburg make skinny tires a poor choice. I would not recommend using very narrow road bike tires east of the Columbia River, although a "fat bike" is not needed. Gravel grinders, mt. bikes, and touring bikes have toured the whole trail.
Trail looks to be gravel. Is it hard packed enough for a road bike? Are some sections more road-bike friendly than others?