Clint's methods on in the field medical stuff wasn't quite what medical training for in the field was actually based off -his own little habits picked up and Natasha's attempts here and there. He'd never actually taken the 'this is how to save your life in a pinch' course that SHIELD offered, mostly because he didn't really want to listen to the medics drone on and on and partly because it was held in the med-lab and he hated that place.
"Okay, whoopie for Dorian, but I'm more looking for antibiotics, fairly standard ones, nothing too specific, unless you have something in the bag of tricks that the hospital provides that clearly states its for post-possession use." None of Rosa's injuries looked to be infected, all of them would scar (yay, trauma) but Clint wanted to at least fire one round of preventative measures into her just in case.
He wasn't sure how Rosa felt about magical healing either (he wasn't entirely a fan, but then he got patched up by robots so who was he to pass comment).