I think that you could say Aragorn would equally feel reluctant to encouarge Eowyn as a lover
and as a young recruit to the war - not as she is a woman, though you cannot deny that in that culture it would be unusual, not appropriate even in some cases, for a woman to go off and fight. But he would not wish to take control over the ward of the King of Rohan, who clearly has another 'commander' she owes her first allegiance to - especially given that Aragorn has not long since been rebuked by Gandalf over his high-handed attitude towards Theoden!
Quote:
Thus Aragorn for the first time in the full light of day beheld Eowyn, Lady of Rohan, and thought her fair, fair and cold, like a morning of pale spring that is not yet come to womanhood. And she now was suddenly aware of him: tall heir of kings, wise with many winters, greycloaked, hiding a power that she yet felt. For a moment still as stone she stood, then turning swfitly she was gone.
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What's clear here is that Aragorn is struck by her beauty, but also her youth, and her coldness. She in turn is struck by his maturity and his power - it's quite a Byronic attraction for her, isn't it?

I'm reminded a bit of the way Jane Eyre views Rochester.
The other interesting thing here is the power dynamic is reversed somewhat - Aragorn is obviously the mature one here, whereas with Arwen, he is very much younger and with less status. It's testament to Aragorn's character and his love for Arwen that he is able to resist the temptation of Eowyn!