I am new to all of this as well but...
I totally understand your need and desire to meet Mary. It would also concern me if I asked to meet someone, anyone, who was in my partners life and they were soooo opposed to it. What's the big deal? Why is she so adamant with her refusal? To me, it almost sounds like she realizes how important it is to you and is using it to exert some control.
I am curious. Again, new to all of this, but traditionally, when we are dating someone, we meet their friends, families, coworkers eventually...its part of knowing someone, it brings you closer. So why would it ne abnormal or wrong to want to meet your partners other romantic partner? Question for those who expressed that he shouldn't want to meet Mary.
Also, the father comments? Completely rude. She is taking the sex and love out of your relationship with Beth with that comment and making you a stuffy, stodgy old man character. Hmmm...maybe that's how she prefers to think of you? Maybe she feels more threatened or insecure than you know and wants to pretend you are just a Platonic fatherly figure in Beth's life rather than a person with feelings and desires just like her.