Dally looks sick because he has just seen that Johnny, the one person he cares about in the whole world, has taken a horrible beating.
When Johnny is beaten by a gang of Socs, the Greasers find him and are stunned at how bad he looks. His "face (is) cut up and bruised and swollen, and there (is) a wide gash from his temple to his cheekbone"; he would carry the scar from that gash all his life. Johnny's white T-shirt is "splattered with blood", and he is lying so still, the boys think he might be dead. Ponyboy thinks that "surely nobody could be beaten like that and live".
When Dally arrives at the scene he takes one look at Johnny and turns away "with a sick expression". Dally had "seen people killed on the streets of New York's West Side", and is no stranger to violence, yet he looks sick now because Johnny is special to him. Despite his brutal home life, Johnny retains an innocence and sense of purity that has managed to penetrate the hard shell Dally has built around himself. Dally cares about few people in the world, but he loves Johnny, and to see someone he truly cares about so badly hurt makes even Dally look sick (Chapter 2).