All considered, I think most folks will be best served with a high contrast light gray surfaced screen with this Sharp projector. On my Firehawk G3, blacks looked very good, very dark. When I watched the same content on my Carada Brilliant White screen (white, gain 1.4), the image was of course brighter, but blacks just weren't real dark, just bright enough that I wasn't quite as pleased with the end result in terms of blacks. Letter boxing on the 106 inch Carada was a dark gray, and somewhat evident. (By comparison, with my Firehawk, and a projector with better black levels, the letterbox areas are barely invisible).
The XV-Z15000 has just barely enough lumens in Movie 2 mode, to fill my 128 inch screen, but that's with a brand new lamp. I would recommend screen sizes up to 110" with typical screens, and you could push it a little larger than that, but 128 inches will be too large when the lamp has a lot of hours on it.
So, look to high contrast gray, or high contrast light gray screens as my best recommendation. Typically the regular HC grays have gains of less than 1.0 and the light grays, 1.1 to 1.3.
There are always exceptions. If you plan a lot of sports and HDTV, and aren't heavy into movie watching, you may well find the plus gain white surfaces like the Carada mentioned to work very well, since for those other types of content, great blacks are not a demand item. Gamers probably would prefer a white surface as well, since many games are very dark.
The Stewart Firehawk G3 is probably too pricey to pair with the XV-Z15000, but look to others like the Da-lite HC Cinemavision, the light gray surfaces from Elite Screens, and others. Carada also has a gray surface screen but it's one of the darker ones, so I wouldn't use the Carada gray unless you are going fairly small in screen size (under 100" diagonal).
Since I mentioned sports and HDTV viewing, keep in mind, when making your decision, that a high contrast gray surface will "reject" ambient lighting coming from the sides, which can really help if you've got some not fully covered windows leaking light in the daytime, that happen to be located on your side walls.
One last thought - a generic one. If you go with a motorized, or pull-down screen, it is important to get one with tensioning, to keep it perfectly flat. Without tensioning you will have small unevenesses, which, unfortunately tend to be very noticeable and annoying when a scene is being panned by the camera.