...is not considered flirting unless other actions or words meant to be
flirtatious accompany it. To some, it is considered rude NOT to look
someone in the eye when they speak to you or acknowledge your presence; to
deliberately avoid the glance of another is often considered the sign of a
guilty conscience. There is no doubt that Eowyn was attracted to Aragorn,
and he was aware of her feelings: "For she is a fair maiden, fairest lady
of a house of queens. And yet I know not how I should speak of her.
When I first looked on her and perceived her unhappiness, it seemed to me that
I saw a white flower standing straight and proud, shapely as a lily, and yet
knew that it was hard, as if wrought by elf-wrights out of steel. Or was
it, maybe, a frost that had turned its sap to ice, and so it stood,
bitter-sweet, still fair to see, but stricken soon to fall and die?"
Strange, if he was flirting, that what Aragorn notices upon looking at her for
the first time is that she is unhappy, and cold. He becomes aware of her
interest in him over the course of their relationship, but he knows that this
is not right -- not only because his heart is Arwen's, but because he knows her
attraction toward him is for the wrong reasons, because, as Faramir puts it,
Eowyn "desired to have the love of the Lord Aragorn. Because he was high
and puissant, and you wished to have renown and glory and to be lifted far
about the mean things that crawl on the earth. And as a great captain may
to a young soldier he seemed to you to be admirable. For so he is, a lord
among men, the greatest that now is. But when he gave you only
understanding and pity, then you desired to have nothing, unless a brave death
in battle."
I think Tolkien makes the situation pretty clear, and to read anything else
into Aragorn's actions toward Eowyn other than politeness and kindness is a
misinterpretation. Had he completely rebuffed and ignored her, he would
have been considered a rude cad, and that simply is not part of his
nature. So he returned her interest with as much kindness and
understanding as he could, and I don't believe even Eowyn mistook it for
flirting. She knew he was not interested in her that way,and was upset
because her interest was not returned, and angry in a way because at that time,
she was not yet capable of accepting his pity for what it was; her heart was
still hard, and it took later events and the influence of Faramir, to finally
thaw
it.