Logically, could the word "University" mean the opposite of "Diversity"?

Ignoring the fact that the word "university" doesn't mean the opposite of "diversity," your logic has a flaw: The "di-versity" comes from the Latin di(s)vers-, which has the same "dis" as in the Latin synonym divertere, namely "apart." Both words mean "to turn away."

"Dy-," meaning two, comes from Greek. "Twice" in Latin is bis.

Somehow, the meaning consisting of one does not imply that it is not diverse.

Indeed, the meaning of universe, which is the one that everything is included in, implies that it has everything and thus it entails a huge diversity.

The original question should be answered in the affirmative: "Universus" literally means "turned into one" "Diversus" literally means "Turned different ways"

Obviously, one thing cannot be both turned into one and yet set in different directions at the same time.

As to whether the universe is diverse, if the answer is "yes" it is because the terms are not being used literally; "universe" merely referring to a large group of things (but not turning them into one thing), and "diverse" merely referring to a multitude of observed properties that things in the large group (universe) possess.