No, he talks to you. Like, if you follow him for a while and daylight comes, he says "So our game ends..." or something like that. Game being you chasing him around Skyrim(he'll say other things too). Ragnar the Red is just a bard's tale, it may not even be true. He just goes back to his grave at Haemar's Shame, if he makes it.
I know that, but thought he also would make a statement near whiterun. Also I know he goes to his grave, what I'm saying is he passes whiterun each time, could be wrong, but I believe he does.