Is there an Halachic obligation from the Shulchan Aruch to say "Shalom Aleichem" when greeting people, rather than saying hello or something of the sort. I heard this might be a Takana from Ezra.

I am unaware of such an obligation, and doubt that there is one. It is certainly not mentioned among the ten Takanot of Ezra (see Bava Kamma 82a). Possibly one of the commentaries on Pirkei Avot(4:~15 or 16, depending on the edition; there is no consistent division of the Mishnayot in Avot) discusses this further- or someone on the 11th chapter of Mesechta Derech Eretz (also called Perek Hashalom). Also Brachot 17a where Rabbi Yochanan Ben Zakai is praised for this attribute. But in none of these places does it necessarily mean more than to be sure to greet someone.