That's always bothered me too. Like, you couldn't even trust Catelyn with this information? Look at how much grief you caused her! Unless the reason he didn't tell her is because then she wouldn't have treated him like crap and that would have looked weird to the outside world, her just being so okay with his…

Is he going to be angry with Ned with he learns the truth? He has spent his whole life internalizing the identity of "bastard" and it's caused him much strife and heartache. Or - did Ned even know that they were married? Maybe Ned just assumed he was a different type of bastard.