The ultimate source is Shabbos 10b, citing Judg. 6:24, ויקרא לו ה' שלום.
For certain purposes it is indeed treated like a bona fide name of Hashem; thus, the Gemara there says (and this is cited as halachah in Shulchan Aruch, Orach Chaim 84:1)) that one may not greet another in the bathhouse with the word "Shalom," just as one may not recite blessings or words of Torah there.
Baer Heitev (:2) and Mishnah Berurah (:6) discuss whether this also applies to calling a person whose name is Shalom. M.B. ends up saying that technically it is permissible, but that "one who fears Heaven should be strict" and change the name somehow, like by eliding the final mem or changing it to a nun. (Say, Shalom, if you read this: what do people call you in the mikvah?)
He also goes on to say that it is best not to write "Shalom" in private letters either, for the same reason. This is further discussed in Shulchan Aruch, Yoreh De'ah 276:13 and the commentaries there; Shach :16 says that most people aren't careful about this, though in Nekudos Hakesef he questions whether it is correct to do so. (Pischei Teshuvah :28 notes that there's a difference in this regard whether one uses the word "Shalom" as a greeting - in which case he indeed presumably intends to call down Hashem's blessings on the other person - vs. when one writes about "shalom" as a concept, like speaking of world peace.)