My theory is that, in the books, when this scene will be revealed, a dying Lyanna will ask her brother to make a bow to protect her child and to not tell it's true nature to anyone until the correct time comes, if it do so.

Ned is a character bound to his honor. So bound to his honor to die by it. So he creates the lie of the bastard and hide the truth from everyone, even it's most close relatives, because he swear to do so.

My theory is that, in the books, when this scene will be revealed, a dying Lyanna will ask her brother to make a vow to protect her child and to not tell it's true nature to anyone until the correct time comes, if it do so.

Ned is a character bound to his honor. So bound to his honor to die by it. So he creates the lie of the bastard and hide the truth from everyone, even it's most close relatives, because he swear to do so.