[The Squatter of Amon Rûdh]

Aragorn didn't pursue Eowyn for a number of reasons: Firstly and most importantly, he was very much in love with Arwen, whereas Eowyn was just someone he met, to whom he meant no harm (re-read the passage in which they meet: he's embarrassed and worried by the infatuation that he sees in her). Secondly, he was betrothed to Arwen, which was once a commitment every bit as binding as marriage (in the Middle Ages, the betrothal was a ceremony in its own right). Thirdly, and closely linked to the above, Aragorn is deeply indebted to Arwen's father, whom he esteems very highly, and to whom he would never consider offering such an insult as the abandonment of his daughter.
Fourthly, Aragorn is sufficiently wise to see Eowyn's infatuation as just that: a passing 'crush' that would end if allowed to do so; leaving her to discover the reality of love later on. We can see that he understands this from his parting words to her on leaving for Minas Tirith.
Fifthly, of course, there's politics: if Aragorn were to break off his engagement to Arwen in Eowyn's favour, the fragile alliance between Men and Elves could collapse utterly. Aragorn has asked a lot of Elrond in asking for his daughter's hand, and to abandon her would be to throw away the greatest gift that his mentor and benefactor could offer him. Whereas Elrond might forgive, I'm not sure that all other Elves would show such wisdom. Theoden, although blameless, could either refuse the match or face the enmity of the Firstborn, becoming the loser in either case.

These things being so, in my view the possibilities include the following:
1: Arwen dies of grief, or becomes an Elven Miss Havisham. Her life is ruined, as is Elrond's. Elrond never speaks to Aragorn again, and the ties between the Elves and the Dunedain are permanently severed.

2: The bond between Rohan and Gondor is temporarily strengthened, but since Eowyn doesn't really love Aragorn, she could fall for Faramir anyway. That situation could fall out in any number of disasters, including a split between Rohan and Gondor and civil war between the King and the Steward. This assumes a victory over Sauron and Faramir's survival, but this would not be a safe assumption because...

3: Aragorn's banner is never finished. He approaches the Pelennor Fields unrecognised and his army is attacked by both sides in the confusion. Eowyn goes with him rather than the Rohirrim, so she's on the wrong part of the field to face the Witch-King. He defeats Minas Tirith, freeing the Nazgul to resume their search, and Sauron's eye returns to Mordor, noticing Frodo and Sam. Game, set and match to Sauron.

Frankly this would be such a drastic turn of events that there's no end to the possible repercussions.

[ August 27, 2002: Message edited by: Squatter of Amon Rudh ]