Damian didn't care that you were older than him, he felt a responsibility to protect you. Because if he couldn't protect the people he cared about, was he really worthy of the mantle of Batman?

Looking at you hurt on a gurney made him feel like a failure. "Are you awake?" Damian asks, trying to hide any trace of concern in his voice, his ego would never let him admit it, but you're his sibling, even if you aren't his blood.