Aragorn might have not been as careful with his life if Arwen had chosen to leave him. He would have gone to battle and fought bravely, since his allegiance to Gondor and the good guys was something separate from his love for Arwen, even the latter motivated him in everything he did. Maybe he would have been even more brave than usual, and reckless, taking too many risks, eventually maybe getting himself killed.
As for what he'd do afterwards, once he returned safe and victorious to Gondor, I don't think he would have married Eowyn. The movie makes a good point in him not encouraging Eowyn even when he thought he'd never see Arwen again. I see him as marrying someone as noble as he is, out of duty for his people, feeling more respect and kindness towards his wife. Maybe someone from Prince Imrahil's kin, someone with Elvish ancestry, to pass on the fertility and creativity represented by this race, that Child of the Seventh Age talked about.
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And no one was ill, and everyone was pleased, except those who had to mow the grass.
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