Firstly, I would like to add to the question:
Rashi on Bereshis 25:20 says that after the akeida, Yitzchak waited to marry Rivkah for 3 years
המתין לה עד שתהא ראויה לביאה שלש שנים ונשאה:
He waited for her until she would be fit for marital relations-three
years-and then married her. — [From Gen. Rabbah 57:1;
From here it's pretty clear that Yitchak knew from the outset that he would/should marry Rivkah.
Similarly, the Rashi that Gershon quoted shows that Avraham knew that Yitzchak was to marry Rivkah.
So back to the question:
Why didn't Avraham tell Eliezer to go get Rivka?
Well, according to the Or HaChaim 24:12 (cited in my answer here)
There was a suspicion that Betuel (and family) would try to foil the prospective match.
We also know that Avraham specially blessed Eliezer before he left (Bereshis 24:7)
הוּא יִשְׁלַח מַלְאָכוֹ לְפָנֶיךָ וְלָקַחְתָּ אִשָּׁה לִבְנִי
So maybe we could say that Avraham decided, that if he would go about the match in the regular way (by telling Eliezer: 'go get Rivkah') - Betuel would foil the match and Eliezer would fail in his mission. So instead he relied on Hashem - ie that Hashem would sort Betuel out [which is what happened in the end] and somehow make the match, which could only properly be done without revealing the identity of the girl to Eliezer.
Edit: Futhermore, the above explanation seems to fit in with Betuel's reaction after hearing Eliezer's story (Bereshit 24:50):
וַיַּעַן לָבָן וּבְתוּאֵל וַיֹּאמְרוּ מֵיהֹוָה יָצָא הַדָּבָר לֹא
נוּכַל דַּבֵּר אֵלֶיךָ רַע אוֹ טוֹב: And Laban and Bethuel answered
and said, "The matter has emanated from the Lord. We cannot speak to
you either bad or good.
We cannot speak to you: to refuse in this matter, either with an unfavorable reply, or with an appropriate reply, because it is obvious
that the matter has emanated from the Lord, according to your words,
that He designated her for you.
.... which seems to imply that it the matter hadn't come from from Hashem - then they would have naturally refused the match - either unfavorably or appropriately - a refusal nonetheless.