Meaning-wise, no. Contrary => opposing some aspect of something. His behavior is contrary to the rules. (against the rules) Contrast = a comparison of differences or the lightness difference between white and black. His anger and her smile showed a telling contrast between their two personalities. (stark difference between two objects in some regard)

It sort-of is, in that both can mean an opposite, and they are often used similarly to describe two sides, or another side of an argument. 'we can contrast the behaviour of the penguins with that of the seals' and 'on the contrary, such people rarely make sensible decisions'. 'contrary' is generally much more definite opposite, whereas 'contrast' more is used to describe, or draw attention, to one difference. But both words have several meanings, so they do not always have this meaning.

You're correct – they both have the Latin prefix "contra" meaning "against." The suffix -ry is used to make a noun / adjective. The "st" in "contrast" is from the Latin "stare" to stay/stand, so contrast means literally to stand against.