I don't think Dawn had anything to do with the decision.
Before Dawn's arrival, Giles has already decided that Buffy doesn't need him anymore. That had nothing to do with Dawn and everything to do with Buffy as a person.
Buffy had been moving further and further away from the ritualistic aspect of being the Slayer, from the mystical side of it, where Giles resides, since S3. S4 and her involvement with the Initiative and their more tactical, practical, military-like take on slaying made Giles feel redundant through much of that season. She didn't need spells or research - she had Willow and Riley for that. Giles was surplus to requirements.
Until the end of the season and 'Restless', when the mystical side of Buffy's Slayer-self reintroduced itself to her. The demon that makes up the Slayer woke up and Buffy's blood lust started churning and she knew she needed help to control that. She needed someone who as trained to train a Slayer. Giles was needed again, Giles was needed to be a Watcher again, legally or not.
In S6, Giles feels like he's done everything he can to help her Slayer side, and his decision to leave is more about helping her human side. It's a very wrong decision, as far as I'm concerned, but I can see how Giles thought it might have worked. Buffy was leaning on him too much, and Giles thought that taking away her crutch would make her stand up straight.
But Dawn never factors in to his decision. He doesn't think about this innocent, scared, fifteen year old child who is left alone with a guardian who is barely hanging on. Forcing Buffy to stand on her who two feet by removing himself from the equation might have worked if Buffy wasn't responsible for Dawn. But the stress of looking after Dawn just adds to the weight on Buffy's shoulders and she buckles under it.
Giles didn't think of Dawn at all.