Arwen's death would be a side effect of Sauron obtaining the Ring, not a direct result of it. If Sauron gets the Ring, everyone is screwed, especially the Elves, because Sauron has a serious hate-boner for Elves. This is never explained in the movies.
Elves in Middle-earth are tragic figures, because they are doomed to grow weary of the world and filled with the accumulated sorrow of thousands of years seeing other creatures live, suffer, and die. Eventually, every Elf will be unable to bear any more heartache, and will depart Middle-earth, sailing to the Undying Lands.
When they were younger, Elrond and his brother Elros, both of whom were half Elf and half human, were given a choice between living as mortal men or living as immortal Elves. Elros chose to be a man and Elrond chose to be an Elf. Thus, Elrond would live forever, but would eventually grow weary and depart to the Undying Lands, never to return.
In the appendices to The Return of the King, we are told that:
... To the children of Elrond a choice was also appointed: to [eventually] pass with him from the circles of the world; or if they remained to become mortal and die in Middle-earth. For Elrond, therefore, all chances of the War of the Ring were fraught with sorrow.
-The Return of the King, Appendix A: "The Numenorean Kings"
Aragorn is a man, so he can't go to the Undying Lands, plus he has king stuff to do, so even if he was allowed to go there, he wouldn't be able to. Thus, because of the "choice" all Elrond's children must make, Arwen has to choose between going with her father to the Undying Lands and living there forever, but leaving behind the man she loves; or staying with Aragorn after her father departs, but becoming mortal.
So it isn't that the Ring is killing her, nor is it that loving Aragorn automatically entails dying. It is simply that staying in Middle-earth means she has to become mortal.
Things get even more complicated by the fact that the Elves have maintained their power through the 3 Rings they possess. As long as Sauron didn't have the One Ring, the Elves could use their Rings with no risk attached. But if Sauron regains the One Ring, the 3 Rings are no longer safe to use; if the One Ring is destroyed, the 3 Rings will become just as useless as regular rings. So no matter what happens to the One Ring, the Elves have to leave Middle-earth. This includes Elrond, of course.
So Arwen is dying because she has chosen to remain in Middle-earth after her father departs. It is less clear why she is literally about to die - in the books, she isn't dying now, she has just decided that at some point in the future she will die. In the movies, she is in the process of becoming dead.
The only way I can make sense of the events in the movies is that Arwen is choosing to die immediately to guilt trip her dad into reforging the Shards of Narsil so Aragorn can become king. Once he does so, she's all like "OMG I GOT BETTER!".
In the books, Elrond doesn't need to be coerced into reforging Narsil - there, unlike the movies, Elrond has always supported Aragorn's journey towards kingship, and Aragorn has always intended to become king. Jackson changed Aragorn a bit- in the movies, Aragorn is "The Reluctant Hero", and has to be pushed and cajoled into doing what he was destined to do. Elrond is a bit annoyed by this, and at one point says something like "Aragorn turned away from that path long ago, he only wants to be a Ranger now".
So in the books, Narsil is remade before the Fellowship departs from Rivendell, but in the movies, Narsil isn't reforged until the middle of the third film. In the books, Elrond told Aragorn he couldn't marry Arwen until he was king, because Elrond didn't want Aragorn to become distracted from his main task of claiming the throne.
In the movies, Arwen actively chooses to die because it will force her dad to reforge Narsil so Aragorn can become the king, and then Elrond finds Aragorn and says "Get off your ass and become the king or Arwen will die". So really, deciding to die was Arwen's way of making sure her dad and her boyfriend did what they had to do.
The line about "Arwen's fate is now tied to the fate of the Ring" makes more sense in this light. Arwen will stay in Middle-earth no matter what, and if Sauron gets the Ring, everyone in Middle-earth, including Arwen, will be boned. If the Ring is destroyed, she'll stay with Aragorn, and die at some unknown time in the future.