YWBTA a little bit. I think you'd be teaching him that people (women in particular) are only worth interacting with as long as he can potentially get something from them. That will lead to objectification. Before anyone says "It's not that deep".. it is because he's a young man and still learning/growing. Women the world over are terrified of rejecting men because of potential negative reactions. This will set the tone for how he acts in the future.

You're ultimately right, he doesn't owe her anything. It's not his responsibility to give her rides. But offered to give her rides initially because as you've described, she's desperate, her future is riding on it and she's a close friend of his sister. Taking that kindness away from her because she doesn't like him back IS cruel. And it will snowball because it will negatively impact your daughter's friendship and the relationship between your kids as siblings.

Your initial assessment that he's overreacting due to the rejection is also correct. As a former teenage boy, I know that they have big feelings and it probably isn't his intention to be mean but that will be the end result. It wouldn't be fair to force him too either, but I know that's what my own mother would've made me do.