I agree absolutely that Aragorn would have never given up in the middle of the war, even if Arwen had suddenly changed her mind.
But I strongly believe his fate could be quite different if he got turned down before the war really started and he had to make his intentions clear. I mean he would have hardly claimed the throne without a good reason. He had been fighting against the Enemy with the Rangers. Besides, his ancestor’s claim for the crown of Gondor had once been rejected (I mean Arvedui the last king of Arnor) and noone dared – or cared – to claim it again. So imho Aragorn wouldn’t make such a claim other.
Anyway, it was Elrond’s idea, as being a King’s father-in-law is better than a vagabond’s