Why didn't Ned Stark support this claim to the throne?

Question

Spoilers for the Season 7 finale of Game of Thrones, "The Dragon and the Wolf".

It has been confirmed in the television series and possibly confirmed in the novels that

Jon Snow is the son of Rhaegar Targaryen and Lyanna Stark.

Additionally it is shown in Game of Thrones that

Rhaegar and Lyanna married in secret and Jon's true identity is Aegon Targaryen VI

Given Ned Stark knew this information why did he choose to support Stannis' claim to the throne instead of revealing the truth about his nephew and supporting him as the heir?

Doing so would essentially invalidate the rebellion which Ned spent years fighting. But this seems less important considering one of the main causes of the rebellion was based on a lie (as Ned knew).

It certainly seems Ned was ready to tell Jon the truth and not having Robert alive anymore would have certainly helped in this.

Answer 1

There's a few problems with this.

Night's Watch

Jon had already gone off to the Night's Watch and sworn his vows. This means he legally could not claim the throne even if he knew about his birth right and wanted to. He swore an oath that he would spend his life at the wall with no titles, and it should be a death sentence to break that oath.

Bastard Inheritance

Even thought Ned knew about Jon's true father, it's likely he did not know about the marriage. This means that in Ned's eyes he is still a bastard regardless and would have very little to no claim for the throne.

Edit: With regards to Warlord099's comment, yes Ned should believe that Jon is legitimate with Lyanna telling Ned his name is "Aegon Targaryen". But! There is an issue of proof then. Even if Ned wants to claim Jon as a true Targaryen he has to prove it which leads to complications that are better left detailed here. Not to mention that even with having testimony or evidence of a marriage, it could be disputed or claimed fake. As we know, the true parentage of a character can be falsely claimed and people can believe it such as Robert's "legitimate" children.

He made a promise

When Ned took baby Jon from his sister he made a promise to protect him from being killed for his father's blood in him. This was mostly to protect him from the new King Robert. But even with Robert out of the picture declaring Jon as King because he is a Targaryen would put him in severe danger from those that don't wish to give up the throne and those that hate the Targaryens and don't want them in power.

Baratheon Line

With Robert's rebellion the line of succession has changed. Robert claimed the throne with a direct line connecting him to the Targaryen dynasty but mainly through right of conquest. This means that the original Targaryen line of succession has changed and is now following Robert's line the same as it did with Aegon after his conquest. This means that even a "known" true born son such as Aegon Targaryen VI from the books or Jon (Aegon Targaryen) Snow of the show would not have a claim without reconquering the kingdom in the name of the Targaryens.

Mix of 1 and 3

As they were both leaving Winterfell for King's Landing and Castle Black, Ned told Jon he was going to tell him the truth the next time they met. This would've been long after Jon had taken his vows and forsaken any claims he could make to the Targaryen house. We can infer that means Ned would be willing to have the truth be known but only when Jon would be safe. For even a true born Targaryen heir at the wall wasn't a threat as Maester Aemon had been there for years. If Ned was insisting on waiting for Jon to be protected by his vows then he would never have gone out of his way to put Jon in direct danger by declaring him the rightful king.

Answer 2

Ned supported the Baratheon rebellion in the first place, being one of the main "captains" of it. Going against that would look kind of daft, to put it bluntly. Also the Targaryens are no longer in power for the Iron Throne that has fallen down to the Baratheons after there successful rebellion. The Targaryens only took power initially from their own conquering.

But that's not the main reason, at this point in the story Jon has already gone off to join the Nights Watch. Let's remind ourselves of their oath:

Night gathers, and now my watch begins. It shall not end until my death. I shall take no wife, hold no lands, father no children. I shall wear no crowns and win no glory. I shall live and die at my post. I am the sword in the darkness. I am the watcher on the walls. I am the shield that guards the realms of men. I pledge my life and honour to the Night's Watch, for this night and all the nights to come

Jon can not become a King due to his oath to the Nights Watch so it would become pointless to support his claim in the first place. It appears that Jon takes his oath after Robert dies, but the timeline isn't clear. Either way though Ned wouldn't know if the oath was made yet or not.

Another reason is who would believe Ned? Yes we have Howland Reed as a source but he's some unknown hermit. Even if Ned told everyone the truth the likelihood of someone believing him is close to 0.

Also remember that Rhaegar's and Lyanna's marriage was a secret known only to even less than Jon's identity so it would be even harder to prove he's not actually just a southern bastard, a Sand. It's not even clear if Ned knew about the marriage in the first place.