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 I think a flash would 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.