FFC's fanatical attention to detail is at work here. Latin males traditionally grow moustaches to denote maturity and machismo, and Sicilians and Mexicans still do. Note that Michael never had a moustache--he wanted to be "legitimate" and "American" (n.b., at the end of the novel, Kay takes vows in the Catholic Church. Michael is disappointed because he wanted his kids to be raised as Protestants--'it was more American").
Slicked-back hair was common among gangsters and tough guys--longer hair, curly hair or parted hair was considered not part of the image.
Also something to denote is that when Fredo is disowned by Michael, in the boat house scene, he still has a mustache. Afterwards, when he's nothing anymore, just a ward of the Corleone's - the gypsy who belonged to no one he always feared he was - at Carmela's funeral, he's cleanshaven. That can't have been an accidental choice on Coppola's part - it's like saying, to be vulgar, he got his balls cut off completely.
Yeah Michael didn't grow one because he didn't want to be seen in any way publicly as an Italian. He has a nice white American wife from a Protestant background, moved to a very non-ethnic state, and honestly, he doesn't seem to have any special love or appreciation for his heritage or culture. He's an Italian-American with an emphasis on the American; he notes his fighting career as a Soldier when defending his reputation, but doesn't consider it an act of persecution because he's Italian. Honestly, it almost seems like Michael has a DISDAIN for Italians. He speaks the language, sure, but the culture seems alien to him, despite having lived there. He's almost a self hating Italian. Like, I feel he may have loved his father, respected him as a man, but viewed his way of doings things as rustic and backward. Michael didn't believe in community, he believed in Michael.
