Dally looks sick because he sees Johnny has been badly beaten by the Socs.
Dally is one of the toughest members of the greasers gang, but he has a soft spot for Johnny, whom Pony describes as the gang’s pet. When Johnny is jumped by greasers, Dally takes it particularly hard. All of the gang members are shaken by it, but Pony finds Dally’s reaction troubling, because he is so tough.
Dally was there, too, swearing under his breath, and turning away with a sick expression on his face. I wondered about it vaguely. Dally had seen people killed on the streets of New York's West Side. Why did he look sick now? (Ch. 2)
Dally cares deeply about Johnny, but he knows that the boy is fragile. He has a difficult home life, because he lives with an abusive and neglectful father. It makes him jumpy. When he is attacked by the Socs, it is almost more than poor Johnny can take.
Johnny was also hurt very badly. Pony says he thought, looking at Johnny, that he might be dead.
Johnny's face was cut up and bruised and swollen, and there was a wide gash from his temple to his cheekbone. … His white T-shirt was splattered with blood. I thought he might be dead; surely nobody could be beaten like that and live. (Ch. 2)
Socs have a tendency to target greasers who walk alone. They prey on the weak. Johnny, like Ponyboy, belongs to the gang because he wants a family. He is not necessarily a tough kid. There is no way he could defend himself against the brutal Soc beating.
After he was jumped, Johnny was even jumpier than usual. It caused things to be more strained between the greasers and the Socs, and led to the fight in the park between Johnny and Bob that caused Johnny to kill Bob in self-defense (and defense of Ponyboy). This is the first link in a chain of events that will alter all of their lives.