I have eye contact with men all throughout the day at work. If I am talking to them, it is impossible not to.
Most men I do not even find attractive. I am not thinking of them in that way and my eye contact means nothing.
I don't think that most men and women are constantly drooling after each other all of the time.
If some man has a problem looking at a woman then if he thinks that works, he should do it. But if a man can control himself so little that he can't even look into a woman's eyes, there is something wrong with him.
I give men more credit than that. I don't think they are all madly oversexed creatures that melt by looking into every women's eyes.
Most people are average looking anyone and don't drive everyone of the opposite sex wild.
I have seen Muslim women where I live, wearing eye makeup. Is that discouraging eye contact.