Sure, I agree. But give them time and space to communicate what they want. Seriously!
Let me try to explain this as you both have no idea and must just like to argue.
Is it not showing a bit of neediness and/or insecurity asking a girl on a second date on the first date?
Its like saying "Hey I'm really insecure and nobody usually likes me or dates me twice so can we lock in the next date now, please?"
Wake up to yourselves.
Take her out, have a good time. At the end of the date tell her you had a good time and ask her to call you some time. Let her initiate contact after the date, even if it takes two weeks. This creates interest on her end and she is confused. "Maybe he didn't like me?, I wonder if he even cares?".
Then when she contacts you invite her on another date. "Hey, its great to hear from you. I'd love to hang out again, when are you free?".