It would be up for her to decide in that case, and the couple together as well. I don't see how you can "go ahead and marry her" if she decides she doesn't want to have to choose you over her family. Depending on the situation it could lead to her being estranged from her family, and not everyone can cope with that. I also think this goes both ways, if you were marrying a guy and his family disapproved for religious reasons he'd have to be the one making that call based on what consequences could come from that... of course, other people could cope just fine, or in other cases the family could just get used to it and accept it eventually.
I don't even understand the thing about asking a woman's family to marry her. Women are not possessions. I think it's a good idea, if both your partner and your family are important to you, to introduce them to each other... but that's just common sense and it's for people no matter their gender. It's not about asking a woman's family to make that sort of decision for her. I also think it's a good idea for people to listen to their families depending on what the subject is, sometimes when they're in love, it's easy to miss flaws the other person might have. For example, my family is very open-minded so I know they wouldn't object to me marrying anyone as long as it was a loving and respectful relationship on both sides. If they had any objections, it would be a good idea to listen to them since it would be because of the other person not treating me right (or vice-versa I suppose). But that's not the same as religious prejudice, however, in a case of religious prejudice, it can be tough for people to be put in a position where they have to choose sides like that.