Simply put, he could have.
There's no "good" reason Holden hits Stradlater with his right hand instead of his left. He does give a reason why the punch with his right hand is ineffective-- he broke it punching windows when he was 13. After it healed, he could never make a tight fist again.
Holden is probably trying to justify his weak performance in the fight with Stradlater, as it is somewhat emasculating. His attempt to defend Jane's honor is pitiful. He can't even make contact with Stradlater's face. Instead, he ends up punching him in the side of the head.
Supporting passages from the text follow.
Holden hits Stradlater with his right hand: "I got up from the bed, like I was going down to the can or something, and then I tried to sock him, with all my might, right smack in the toothbrush, so it would split his goddam throat open. Only, I missed. I didn't connect. All I did was sort of get him on the side of the head or something. It probably hurt him a little bit, but not as much as I wanted. It probably would've hurt him a lot, but I did it with my right hand, and I can't make a good fist with that hand. On account of that injury I told you about."
The "injury" he's referring to occurs the day his younger brother Allie dies: "I was only thirteen, and they were going to have me psychoanalyzed and all, because I broke all the windows in the garage. I don't blame them. I really don't. I slept in the garage the night he died, and I broke all the goddam windows with my fist, just for the hell of it. I even tried to break all the windows on the station wagon we had that summer, but my hand was already broken and everything by that time, and I couldn't do it. It was a very stupid thing to do, I'll admit, but I hardly didn't even know I was doing it, and you didn't know Allie. My hand still hurts me once in a while when it rains and all, and I can't make a real fist any more-- not a tight one, I mean--but outside of that I don't care much"