Green’s Dictionary of Slang

1. (US juv.) to humiliate/initiate a boy by smearing his penis with some substance, urinating on him, hitting his penis with a knotted handkerchief etc.

1935		S. Kingsley Dead End Act I: Yuh li’l sissy! Let’s cockalize him!
1975		G. Legman No Laughing Matter 111: The boy’s penis is also sometimes struck with a knotted handkerchief, or smeared with paint, or is urinated upon by one or all of the other boys, under the name of ‘cockalizing’ [...] him [...] The humiliating intention of such forms of ‘cockalizing’ is perfectly evident.

2. (US) to beat, to defeat heavily.

1967

(con. 1940s) M. Dibner Admiral (1968) 338: No matter what you said, they’d cockalize you.