I suspect Jon would fight the White Walkers until the bitter end rather than become King because he lost the wall. That would be like taking a reward for losing, and it doesn't strike me as part of his nature.

However, if he were to somehow either defeat the White Walkers entirely, or come up with a suitable truce system, then I could imagine him becoming King.

The more I watch Tommen, the more I think it would be okay if he just remained King. I can see him eventually smartening up to those around him.