Can a Jew name a child after a (deceased) non-Jew (non-relative)? Is there any prohibition against it?
There appears to be no formalized halachic restriction on how to name a child. One commonality is not to name a child after a Jewish enemy, but I know of no halacha against doing so. (It would be weird to find a Jewish boy named Nebuchadnezzar :-) Besides it being a "nasty" name, it's too long. And, you wouldn't want to shorten it to "nebbish".)
Consider these examples:
Abram and Sarai were non-Jewish names. Is Yishma'el a "Jewish" name? Yet, we know of a Rabbi Yishma'el who is mentioned many times in the Talmud, as well as our morning davening.
Names such as Dov-Ber, etc. are named after animals. I know a few people having this name who were not named after anyone.
As @sabbahillel indicated, above, Alex(ander) is a popular name. Many German Jews named their sons Franz after Franz Joseph.