The thing is girls won't voluntarily meet your eye except maybe in social circle, they don't want to be caught checking you out so they'll maybe check you out if you appear occupied, and then look away, obviously there are exceptions but this is pretty much the way it goes for me, basically girls are very good at stuff we have to work at, like posture or eye contact discipline or making guys chase, because these are the skills they need to get through the day without being hassled/judged and remaining in control. Guys don't have such issues. And while some girls are outgoing/friendly and will meet your eye on public transport, chat or whatever they are a minority.
So what you have to do is be much more decisive in your approach, if they check you out and look away, then instead of trying to catch their eye and proceeding from there, go over to them and engage them in a way that can't be ignored, maybe not your whole body (too intimidating) but get right next to them and say in a strong voice "hey there" or whatever opener you want to use, if they still don't acknowledge you then say "can you hear me?" or whatever. Because they may test you, but you know they're attracted cos they checked you out, so you can engage them confidently.
So in summary the issue might be your expectations re eye contact. Of course I might have misunderstood what the issue is but I hope that helps

go get em bro