A black and white interpretation of this scene would simply be that it was unfair for Arizona to ask Callie to promise such a thing and that it was wrong for Callie to promise such a thing.
But Arizona was desperate, scared and running out of hope. She was grasping for something. Anything. Any kind of reassurance. So she took it from Callie. She was inconsolable. Falling apart. Callie saw the need. So she gave her what she needed in that moment. Told her what she needed to hear. As a way to comfort her.
And the relief that washes over Arizona is proof of how much weight she put on that promise. Maybe Callie should have added a caveat; "I promise, unless it’s the only way to save your life." But Arizona probably wouldn’t have felt as relieved being faced with the reality of dying on top of this. And maybe Callie assumed it was implied or understood.