Aragorn makes eye contact out of respect, as he would do with Theoden, Eomer or anyone else. Aragorn is indeed aware of Eowyn's feelings for him, and does an excellent job of showing respect and concern for her feelings without leading her on. He even refers indirectly to Arwen in his confrontation with Eowyn before he takes the Paths of the Dead: "Were I to go where my heart dwells, far in the North I would now be wandering in the fair valley of Rivendell."

Later, when he heals Eowyn after her fight with the Witch-king, he leaves the room before she wakes up, so as not to give her the wrong impression. He is grieved by her hurt and sorrow and feels the need to make things right:
"I saw also what you saw, Eomer. Few other griefs amid the ill chances of this world have more bitterness and shame for a man's heart than to behold the love of a lady so fair and brave that cannot be returned. Sorrow and pity have followed me ever since I left her desperate in Dunharrow and rode to the Paths of the Dead, and no fear upon that way was so present as the fear for what might befall her. And yet, Eomer, I say to you that she loves you more truly than me; for you she loves and knows; but in me she loves only a shadow and a thought: a hope of glory and great deeds, and lands far from the fields of Rohan."

*******************************************************
Although now long estranged, Man is not wholly lost nor wholly changed.
Dis-graced he may be, yet is not de-throned,
and keeps the rags of lordship once he owned:
Man, Sub-creator, the refracted Light
through whom is splintered from a single White
to many hues, and endlessly combined
in living shpes that move from mind to mind.
Though all the crannies of the world we filled
with Elves and Goblins, though we dared to build
Gods and their houses out of dark and light,
and sowed the seed of dragons--'twas our right
(used or misused). That right has not decayed:
We make still by the law in which we're made!