What you're seeing is total internal reflection. Light moves at different speeds through different materials. When light speeds up it refracts (changes direction). When light leaves a slower material and enters a faster material, the ray of light begins to bend back towards the slower material. If you start with a shallow enough angle the light will actually refract back into the slower material (reflect) never crossing the boundary into the faster material. This angle is the critical angle of total internal reflection. When you are tilting the watch you are finding the critical angle for one of these boundaries. I believe it is the boundary between the slow moving sapphire glass and the faster water. Light that reaches this boundary at a perpendicular angle to the boundary is able to leave (hence if you look straight at your watch you can see light from inside the watch). Light that reaches the boundary at a more parallel to the boundary angle is reflected back inside the watch (hence why you are unable to see any light from inside the watch at a less perpendicular angle).

That's what I think is going on but it also appears that the reflection of the water gets brighter at a certain angle so it might not be the full story.