Dude is gender neutral.
Chick comes from "chica", which by itself comes from the Latin "ciccum" as in pea.
By calling a woman a chick, you are called her a "chick-pea".
There's even some association with "chick bird", as in a bird you hunt.
Modern usage associates the use of "chick" with the beauty of the individual. Calling a woman a "chick" is using her physical appearance to describe her, disregarding her actual skills and achievements.
By calling a woman a chick, you are either:
A) Calling her a young girl (women are not children)
B) Calling her a young bird (women are not prey)
C) Calling her a young plant (women are more than brainless beautiful things)
All of those can offend.
The equivalent would be calling a men "boy toy" or some such slang. Not something you can do without earning a punch in the face.
Derogate: –verb (used with object) 3. to disparage or belittle.