How was that his biggest mistake? Robb had no claim to the Iron Throne. The North rallied so strongly behind him because they wanted to be their own kingdom again, not subjects to a foreigner in the distant south. They had no interest in remaining united with the southern kingdoms, and Robb had no interest in ruling them.
Besides, as it was, if he'd won the war he likely would have ended up in control of the North and the Riverlands, and possibly even the Vale of Arryn if his aunt decided to declare for him. That would have been the majority of Westeros anyway.