But wouldn't talk about his mother, and didn't prevent him from signing in the Night's Watch. Also never even talked of legitimizing him.
If we believe R+L=J, then that was too dangerous. Plus he said he would the next time they met, he just succumbed to Sean Bean syndrome.
The Night's Watch is seen as a thing of honor and stuff in the North. His own brother went there, and it's a Stark tradition to have a family member go to the Wall every generation.
And if he legitimized him, he would've been a threat to Robb in the succession thing, Catelyn would've knifed him in his sleep or something ridiculous, plus he'd piss off the Tully family.