Would Robb kill Jon

[Lady Blizzardborn]

Robb would not have killed Jon. He would have absolved him of the vow. As King he would have that right. Of course if Jon didn't keep Robb from marrying Jeyne Westerling, both boys would have died at the Red Wedding anyway.

[Pod The Impaler]

Robb would have to find some excuse and march him back to the Wall.

"Doubtless Jon came all this way to cut my meat for me. I keep forgetting how, it's really quite embarassing."

That, and he'd have to send a few dozen prisoners to the Watch as recruits, along with Jon, as an unofficial apology.

[averde]

Robb would have been in a pretty damning position if Jon had deserted. Either execute him and be a Kinslayer, or spare him and make an enemy out of the Night's Watch.

I think making an enemy of the Nights Watch is probably not a concern of many lords, especially one who's fighting a massive war.

Upset Nights Watch < let brother live.

Not even a contest.

[The Wolf Lord's Daughter]

Of course not. Robb was willing to break an oath to a person who was known not to be trustworthy just to save Jeyne's honour so he'd be willing to step on a few toes to save his favorite brother and best friend. If Jon had reached him, he'd first try to convince Jon to go back to the Wall or to the Free Cities and failing that, he would've given 100 men or so to the watch in exchange for his brother and absolved him of his vows.

[Winterfell Resident]

No.

Robb would've have probably told him to go back to the wall and to fulfill his duties as a man of the nights watch.

He would've overlooked it.

And if memory serves me right it would be up to the Warden of the North to decide punishment. I believe since Ned Stark was imprisoned at the time that responsibility should have fell to Robb Stark.

So my answer is no.