He should have left the Kelvin timeline, too. We don't know that there wasn't something he was supposed to do in the Prime timeline that he didn't do because he was no longer there to do it.
That doesn't make sense. It's not like someone went back and removed him from a point in the timeline where he'd been before; he was no more "destined" to do something after he left than, say, Kirk would've been destined to do something after he died on Veridian III.
Also, he didn't have the means to leave the Kelvin Timeline. The Jellyfish and its red matter were destroyed, and 23rd-century Starfleet didn't have time travel or the capability of crossing between timelines. Even if they did, Spock wouldn't necessarily have been able to find the Prime timeline.
More to the point, Spock felt an obligation to help the survivors from Vulcan's destruction. They needed him more than anyone in Prime did. And he felt responsible for what happened, because it was his failure to save Romulus (Prime) that drove Nero to take revenge by destroying Vulcan (Kelvin). So of course he wasn't going to abandon the Vulcan refugees. Helping them was what he was "supposed to do."