Another kosher animal mentioned in our parasha (Re'eh) is the tzvi צבי - gazelle. ("Gazelle" also has Semitic roots, coming from the Arabic ghazal, but other than this guess, I haven't found any Hebrew cognate.) Is there a connection between the gazelle and the Land of Israel - as in the phrase Eretz HaTzvi?
Horowitz, in his section on how some Hebrew consonants are actually two different letters writes that tzvi - "delight, ornament" and tzvi - "gazelle" are not related. He writes:
For centuries scholars, believing there was only one root here, connected the two meanings by saying that the gazelle was a thing of beauty, a delight. But actually these two words are from different roots; the צ 's are different.
Klein also shows two different roots. For the tzvi meaning gazelle, he writes:
Related to Aram-Syr טביא, Arabic zaby, Akka. sabitu (= gazelle).
He also mentions that this is the origin of the name Tabitha.
As for the meaning of "beauty", he connects it to the root צבה - meaning "to wish, desire". This verb is found in Aramaic Daniel 6:18, in the Aramaic translations to Biblical Hebrew words such as חשק, חפץ and רצון (all meaning will or desire), and in the Talmud as well (Yoma 86b, 87a). Therefore a translation of Eretz HaTzvi could be "a desirable land", which would pair up well with the phrase ארץ חמדה - Eretz Hemda, which means the same thing.
From this root we also get the Hebrew word צביון tzivyon, which originally meant "will or desire", later became "beauty", and in Modern Hebrew means "character, nature".
However, all that said, I don't think we need to entirely disconnect "gazelle" from "beauty". While they come from different roots etymologically, the gazelle is certainly a beautiful animal. In fact, in Shir HaShirim it is used often as a metaphor for the beloved. I would assume that Shir HaShirim was using two commonly known words in a poetic way to make an association between one another.