The answer that Rav Hershel Schachter quoted to us many years ago in a Chumash shiur was that the Targum Yerushalmi translates "Vayekhal Elokim bayom ha-shevi'i.." along the lines of "Nikhsefa v'gam kalta nafshi," to mean that Hashem desired the day of Shabbat.
The above answer, along with several others, is cited by Rabbi Shmuel Shprecher in an artilcle reprinted in Yovel Shai, collected essays (Hebrew, Jerusalem, 1999, privately printed) Among the other answers: 1. Karata translated as "designated," rather than "called." 2. Group the words "v'kedashto hemdat yamim," (then insert comma), thus removing the need to look for where G-d calls Shabbat "hemdat yamim.
Rabbi Shprecher's own suggestion: "oto" to be translated as "its sign," referring back to Ex. 31:17 that calls Shabbat the "ot" between Hashem and the Jewish people. He translates: U'va-shevi'i (with the gift of Shabbat) Ratzita bo (you have chosen the nation of Israel) v'kidashto (and sanctified that nation). Chemdat yamim (the paytan's own term) oto karata (you called its--the nation's--sign) zekher l'ma'aseh v'reishit (indeed that verse connects Shabbat to ma'aseh v'reishit.)
Of course, only the first answer finds a way of attributing the hemda, desire, to Hashem through an actual source connected to a verse.
If you like, I can scan and email you the entire article.
Moshe Rosenberg
SAR Academy
