Assuming the projector is connected as the second display device, then the following will work:
var projector =
DisplayDevice.GetDisplay(DisplayIndex.Second) ??
DisplayDevice.GetDisplay(DisplayIndex.Default); // in case the projector is unplugged
var gw = new GameWindow(
projector.Width,
projector.Height,
GraphicsMode.Default,
"My Window",
GameWindowFlags.Fullscreen,
projector);
If you have more than two monitors, you will have to iterate on DisplayDevice.GetDisplay()
until you discover the projector.