Lets break this down into sub-questions to make the answer more obvious:
It is possible for someone NEVER to use the flash?
Yes. Buy a DSLR without a built-in one and do not pay for an add-on flash, and voila!
Can you make great photos without EVER using a flash?
Yes. Just look at photos taken without a flash. My entire gallery has been taken without a flash. Sometimes I use flash, it is extremely rare but not an absolute for me.
Are you going to run out of photos to take if you do not use a flash?
No. There is always more to shoot, more subjects, more angles, etc.
Can you take every type of imaginable photograph without a flash?
No. Some shots require flash to get it a certain way. High-speed photography like water-drops are a good example.
Can a photographer be a professional if he only shoots photos without a flash?
Why not? It is part of style. Some photographers are even more restrictive. I've heard of one which shot everything in natural light with a 50mm lens over a 30+ year career. He neither run out of photos to take nor of people willing to pay for them.
Honestly your friend probably has an excellent advantage on his side. His wedding photos probably look unlike most typical wedding photos. That is a pretty cool selling point!