Why would he tell the one person who most hates Jon a secret that could cost Jon his life? Robert's approval of the murders of Rhaegar's legitimate children would hardly permit any sensible person to reveal the knowledge of a bastard child of the same father. Catelyn's morality is largely governed by the approval of her more just husband and son - given the opportunity to clear Ned's reputation of a stain that also embarrasses her, plus remove the slightest possible threat to Robb's inheritance, she'd spill the beans in a heartbeat.

For example, if the Kingdom in the North had lasted a bit longer, Robb's legitimizing of Jon might have actually become an issue, and had Catelyn known his true parentage, she'd have blabbed in a second. Before Ned's death, she might have arranged for word to reach court to bring Robert's hatred down on Jon and arrange for his exile or removal from Winterfell at the very least!

While the laws of the Seven Kingdoms might preclude Jon inheriting Winterfell or usurping that inheritance from one of Catelyn's children, as a practical matter, it should be clear by now to readers that, right or wrong, what happens is what you can get away with. Imagine something happens to Robb & and Ned dies when Jon is a young man, before Bran or Rickon is really old or mature enough to inherit, and Catelyn has found socially desirable southron husbands for her daughters. In such a scenario, Jon might have a good chance of taking Winterfell. His being raised as Ned's son in all but name would be a strong factor in the minds of the northmen, and certainly make him a preferable choice to being ruled by a boy they can't trust yet, or a daughter & her husband "from some flowery seat in Highgarden or Dorne." Jon would be a better-known quantity, would appear qualified and a true Stark, unlike the Tully woman with her sept to the Seven, and her Tully-by-appearance children and other southron ways. Even if he was not powerful enough or inclined to challenge his "siblings' " inheritance, others with real power (Boltons, for instance) might seek to use him so. And how would he come into contact with potential allies of this nature, how would he become known to the lords of the North and gain approval as a possible candidate to take over the North? By living in Winterfell as a son of Ned Stark. That was the real source of her resentment, and an issue that could have been easily remedied by her exposing the truth of his parentage, no matter the cost to Jon or Lyanna's reputation.

If Ned can love Jon because he is the son of his sister, how many more can hate him because he is the son of their enemy? "The Sworn Sword" & "The Mystery Knight" should make it plain what trouble a scion of the Targaryens, no matter the circumstances of his birth can cause the kingdom. Some, of the Varys & Pycelle mindsets, might have him assassinated as a boy, just to prevent this sort of problem, and never mind those like Robert. Even Catelyn's father, who was ruthless enough to force an abortion on his own daughter in order to create a better alliance would probably be willing to have something done to the son of the man who cost him a son-in-law and who threatened, even indirectly, his grandsons' inheritance of one of the great seats of the Seven Kingdoms.

Setting aside the issues of Ned's honor and promises, there was no reason to reveal this information. No doubt this was why Lyanna extracted the promise from him, if indeed she did.