My feeling is that Aragorn's heart was already given to Arwen. The attraction was one way, Eowyn attracted to Aragorn. JRRT wrote that Aragorn saw her plight and pitied her, but that was it. If Aragorn had betrayed his betrothal to Arwen, I think he would have betrayed the entire cause of the Free Peoples and their budding relationship would have been swallowed up by Sauron's war machine.
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