It all depends. While it is true that the physical medium of fiber optic cables would be unaffected, that fiber has to be converted back to electrical somewhere with a media converter to be usable as a means of data transmission. Fiber has media converters on each end, even if it's just a GBIC / SFP and not a full media converter box. All of those pieces require electricity to operate and are themselves static sensitive devices that can be fried with a large enough EMP. If that incident on Christmas in Tennessee would have brought that building down it would have done a number on a sizeable chunk of the east coasts connectivity to the rest of the internet for some time. Service providers could have routed traffic through other equipment, but, imagine a scenario where that other equipment were also fried.

I've never put this in text but everyone should be thinking about doing the following: build or create a type of faraday cage. If you can dig a deep enough hole or place equipment deep enough underground it will survive an EMP. Get several HAM and CB radios, solar generator gear, look into doing IP over CB/HAM, and maybe consider newer mesh technologies like locha mesh. This stuff is (relatively) cheap and would enable you to broadcast information even if the whole power and communications infrastructure goes down. Problem is, you'll need to find at least one other person to do the same, otherwise you'll just be broadcasting to the ether.