The secession of the North from the Seven Kingdoms under the purported government of ‘King’ Robb Stark I was almost certainly illegal under public international law.
International law has consistently disfavoured impairing the territorial integrity of states (or ‘Kingdoms’).1 Secession (and the consequent violation of the sovereignty and territorial integrity of the Seven Kingdoms) can only be justified where a ‘people’ have had their right of ‘internal self-determination’ so grievously violated that the only remedy is external self-determination (i.e. creating their own state). 2
The citizens of the North may constitute a ‘people’. They possess different religious customs to the South in their worship of the Old Gods, and different ethnic characteristics, being descended from the First Men rather than the Andal.
However, their claims of violated self-determination are insubstantial. Given the inherently feudal nature of the Westerosi state, the North has a great deal of autonomy over its own cultural, religious and governmental affairs, and is led by its own Northern leaders, chosen according to Northern custom. Whilst the political assassination of former Northern leader Eddard Stark was possibly unjust, it is hardly equivalent to the ethnic cleansing, human rights abuses and massive political upheaval that has characterized successful secessions.3
Moreover, before the North could lawfully secede, all measures short of secession to address the violated self-determination need to have been attempted. Whilst peaceful negotiation with the Lannister puppet government was likely impossible, Stark had yet to enter negotiations with the lawful Westerosi government of King Stannis Baratheon I, when he unilaterally declared independence.
Finally, even if the North under Stark had a valid right to secede, it is questionable whether he even created an effective state under international customary law.4 A state requires an ‘effective’ government, i.e., a government capable of exerting control through the maintenance of law and order over the state’s territory at the exclusion of foreign military forces.5 Given that the North’s capital, Winterfell, was burned to the ground by the foreign occupiers, the Ironborn, this probably precludes Stark’s government (which is being run from his campaign tent in the Riverlands) from having effective control over his territory and thus from having achieved statehood from a successful secession.
1. This was noted in the Separate Opinion of Judge Yusuf in the 2010 Accordance with International Law of the Unilateral Declaration of Independence in Respect of Kosovo (Advisory Opinion) of the International Court of Justice.
2. As stated at  of the 1995 decision of the African Commission on Human and Peoples’ Rights, Katangese Peoples’ Congress v Zaire.
3. In its 1998 judgment Reference re Secession of Quebec.
4. According to the Montevideo Convention on the Rights and Duties of States (1933) which forms part of international customary law.
5. As noted in The Aaland Islands Question, a 2010 report of the International Committee of Jurists.