Why did Jon knock Ygritte down instead of letting her shoot redbeard with an arrow?
He was outnumbered. He was probably the best young swordsman on the Watch because of his training as a Stark, under real knights. But he wasn't a Barristan Selmy type swordsman. I suspect he saw the act as saving her life. And not having her fall to his fate because it was clear she would defend him.
If he takes her with him, the best she will ever be is a prisoner. Or a target by the Wildings. Leaving her, she has a chance to consider what he said and desert, or if the opportunity arose, pretend to be someone else and be spared. Clearly if he himself would be hesitant to kill a young girl, so would others.
It is hard to see what Ygritte would like in Jon other than his looks because Kit Harrington is IMHO an atrocious actor. He's banging the actress in real life and he still can't generate any chemistry with her from his end.