Nothing wrong with noticing, but discretion is the better part of valor. The following is my algorithm for casual social interactions with total or near-total strangers. It’s kind of long, but may be of interest to any guys uncertain what to do in public.
When I’m in public, I tend to notice both men and women, both because I appreciate the human body, and because I’m interested in clothing styles, and because I’m very observant due to decades of riding bikes and bicycles and cultivate it as a survival skill, and because I never really completely trust strangers and keep a weather eye for threats or even just someone carelessly bumping me. And I notice children because they’re less careful and I don’t want to accidentally run into one and a small corner of my mind is always a little concerned for kids in the vicinity of vehicles. I even notice pets because animals are awesome and also higher risk of being hit. And I notice women who are attractive to me as a man, though this tends to be lowest in my attention simply because my natural vigilance doesn’t leave a lot of room in my head for other things, and what attention I do have left is usually already engaged in thoughts not about the here and now.
But what I don’t do is stare, at anyone, or their pets, or their kids, or their cars, or their clothes. By applying the staring is rude rule in general, I don’t have to worry about staring at any one person and making them uncomfortable.
One advantage to my little long-since automatic alertness is that I occasionally catch someone staring at me.
Half the time it’s a young child or baby, in which case I give a quarter-second formal friendly smile so they don’t feel threatened and then move on with my half-conscious environ scanning; I figure kids don’t know any better and have the right to stare.
4 out of the other 5 times it’s a man. I think what usually happens in these cases is that the man was casually glancing but locked eyes with mine and didn’t want to be the first to look away (humans have some weird ideas about dominance), so I’ll give them a nod or a Spock eyebrow or a casual salute depending on my mood and what I read in them (nod for friendly glance, but if I see any hostility in their look a lightly humorous gesture lets them know I’m not playing that game).
About one time in ten it’s a woman. I think what usually happens in these cases is that they too were casually glancing around and locked eyes. But the social power dynamic is very different. So I give a quarter-second smile, but one tailored to convey peer dignity, possibly with a slight nod of acknowledgement, then look away. Because women are targeted far more than men by (almost always if not always male) predators, they rationally have much more to worry about. Ergo, it makes sense for men to, when eyes meet, give the most curtly and neutrally polite acknowledgement possible and move on. Snapping eyes away without acknowledgement may increase the estimate of danger. Holding eyes is creepy.
If it seems strange to have worked out such a detailed algorithm consider that, A, I’m HFA and have approached social situations as analytical problems since childhood when I first realized I did not understand them instinctually and, B, I design algorithms for a living.
ETA: As for how to confront the issue of violence against women, I think Patrick Stewart has a good idea about the bare minimum a good man must at least do.
