On page 118 of my edition, Ralph and Jack are getting into an argument about looking for "the beast," which many of the boys on the island now believe exist. Here's the exchange they have:
He [Ralph] took a step and halted.
"Wait a minute though! Where does the pig-run go to?"
"The mountain," said Jack, "I told you." He sneered. "Don't you want to go to the mountain?"
Ralph sighed, sensing the rising antagonism, understanding that this was how Jack felt as soon as he ceased to lead.
"I was thinking of the light. We'll be stumbling about."
"We were going to look for the beast."
"There won't be enough light."
"I don't mind going," said Jack hotly. "I'll go when we get there. Won't you? Would you rather go back to the shelters and tell Piggy?"
Now it was Ralph's turn to flush but he spoke despairingly, out of the new understanding that Piggy had given him.
"Why do you hate me?"
The boys stirred uneasily, as though something indecent had been said. The silence lengthened.
Ralph, still hot and hurt, turned away first.
So Ralph asks Jack the embarassing question, "Why do you hate me?", which Jacks leaves unanswered. It's embarassing because it points out that Ralph knows how Jack feels about Jack not being leader, and that Jack's actions on the island and his attempts to gain power haven't gone unnoticed. It also points out that Jack has emotions and can be hurt too, when he's tried to portray himself otherwise (Note the way that Jack insults Ralph by mocking Ralph's friendship with Piggy.)
After Jack mocks Ralph for suggesting that they head back to the littluns and Piggy, Ralph asks, "Why do you hate me?"
What is awkward about his question is that Ralph has been feeling a dislike towards Jack for some time now. Jack wants to lead. He wants to hunt, and he wants to kill the beast. Ralph sees these qualities in him. He sees that Jack is savage underneath it all, while earlier we read that Ralph was daydreaming about being back home in his cottage, all cleaned up with trimmed hair and trimmed nails. He longs for the civility, so he feels the contrast with Jack. When he asks this question, he has made it clear that they don't get along and they have different goals. It is awkward for them both, since they both are leaders. Ralph is the elected leader, but Jack is the natural leader. The phrase "this island isn't big enough for the two of us" comes into play here. Only one can lead, so destruction is inevitable.
It is embarrassing because it is the first time it has been brought up--and it's brought up in front of others. Neither wants to show any weakness if they want to control the others. After asking it, Ralph blushes and Jack broods. Both show emotion even though no answer is issued.