Weddings are a special type of engagement in my opinion. The rule I have always heard is "Flash kills the romance". It is usually a truth in wedding photography.
I rarely use a flash in my own wedding photography, but it does come into play in certain situations. I think it would be completely acceptable not to ever use a flash if that is the effect you are going for. If you shoot with prime lenses and wide open apertures, you really do not need to use a flash to freeze your subjects even in dim light.
Where would a flash be useful?
-Posed group shots indoors, such as in a church.
-Indoor wedding reception at night
-Dimly lit church procession
All of these situations can be covered if you shoot with a wide open aperture such as f/1.2 and a high ISO, but you are going to be hit with a razor thin depth of field which might not be adequate.
Overall, I would say it is unlikely, but possible that a professional wedding photographer would never use any type of flash/strobe/lighting besides natural light.