I recently saw a post on facebook written by a friend of a friend (in my feed because the mutual friend had commented): a man, commenting on a couple situations in which he'd recently seen sexism in action, concluded with "it's time to stop acting like women belong to you". How awesome! There are many, many, men who don't comprehend that. I know next to nothing about this original person, he's apparently trying to be a good guy - but even while trying to point it out, the internalized sexism runs deep.
For one of his examples, it totally fit within this concept - it was about a guy thinking his opinion of a women's appearance is important enough to override her own preferences. That's basically making her body public property, and is not ok, and is about men thinking that women belong to them, that they have an inherent right to make decisions for or influence women.
The other situation, he came to the same conclusion - he witnessed behavior that he thought indicated another man believing he had a right to a women, and that that guy should have acted differently. I, however, read it quite the opposite. A guy (a customer/client) gave a female employee his phone number, and suggested they hang out. The original poster (the one trying to identify sexism), proposed that it would have been more respectful to first find out if she was "available or interested", and specifically pointed out that he hadn't even found out if she was single.
The first step to asking out a potential date should be finding out if she's single and available? No. What does it ultimately boil down to if you're finding out if a woman's single? Whether she already belongs to another man. That's not respectful, that's just another iteration of sexist behavior.
This is the same sexist belief that leads to women wearing fake wedding rings or telling a guy in a bar "sorry, I have a boyfriend" in order to prevent being hit on, because saying you're not interested won't do anything - only saying there's already another man involved will make him back off.
It's about a guy believing he has an inherent right to talk a woman into giving him a chance if she doesn't already belong to someone else. You're not entitled to a woman just because she's not taken by another man yet. You're not entitled to a woman at all. If you're given a chance, it's by her choice. Her being interested in you (and giving her your phone number and asking to text you if she wants seems like a reasonable enough way to find that out to me) is the first hurdle; whether logistics make it possible to take it anywhere is - or, rather, should be - entirely secondary.
I'm not saying I get hit on all that often. But the last instance I recall (and what I feel like I see frequently, in real life and in media), it took the form of "Can I ask - are you married?" When I answered yes, he said, "hey I had to give it a shot, have a good day." He gave it a shot? What? He had absolutely no idea if I was interested in him whatsoever - but apparently, giving it a shot only required seeing if I was single. Presumably, if I was single then surely he could talk me into giving him a shot.
If a woman's interested, then it's her choice to determine if she's available to her interest - whether that's by being single, becoming single, cheating, whatever. (Maybe you're not interested in being the other man - but that only becomes relevant if she's interested. If she's not interested, then you won't be cheating with her anyway.) It's on her. But if she's not interested, don't try to convince her. Don't think you have a right to convince her. If you think you have a shot so long as she's single, you're assuming women belong to you.