Éowyn mistakes her hope in Aragorn for love. This is perfectly understandable, since Éowyn has very limited experience with such a thing as love. To her, the love for such an obvious leader as Aragorn is valid because she has known no other kind of love, and also because she has had to remain helpless and ineffectual against her kingdom's enemies.
In this way, love becomes synonymous with power. Again, this is perfectly understandable for someone who’s spent her youth feeling powerless.
The fact that Éowyn is a woman is almost moot. Any soldier, as is later suggested in “The Steward and the King,” would love Aragorn for the exact same reasons as Éowyn. In fact, her feminine instincts of love are pushed aside completely (or perhaps simply replaced) in favor of admiring and following Aragorn.
What’s more interesting are Aragorn’s reactions to Éowyn. He seems to display hints of interest, or at least of intrigue, upon initially meeting her. But his own obligations to both Gondor (as heir to the throne) and Arwen (his future bride) make it impossible for him to return any sign of favor.
Though Aragorn comes across as yet another man limiting Éowyn’s ability to do anything to help her own people, he is the catalyst that compels her to ride into battle disguised as a man. Were it not for this--and we can imagine that Aragorn saw such action as inevitable--the outcome of The Battle of the Pelennor Fields might have been drastically different.
After the War of the Ring is over, Aragorn blesses the marriage between Éowyn and Faramir. And his quote on the matter sums up nicely his feelings all along: "I have wished thee joy ever since first I saw thee. It heals my heart to see thee now in bliss."