The answer to this question can be found in Chapter 30 of the book.  Briefly, Jim will not agree to "run for it" with the doctor because he has promised Long John Silver that he will not.  Jim does not feel that it would be right for him to go back on his word.

Jim feels that he owes it to Silver to keep his word.  Earlier, Silver has really stuck his neck out for Jim.  He has persuaded the other pirates not to kill Jim when Jim falls into their hands.  Not surprisingly, that makes Jim feel like he should play fair with Silver.  Before he is allowed to talk to the doctor, Jim has to swear that he won't run away.  Because he owes Silver, and because he is an honest person, Jim refuses to go back on his oath.

In Treasure Island, why doesn't Jim agree to "run for it" with Dr. Livesey?

Young Jim Hawkins, the narrator of Robert Louis Stevenson's adventure classic, Treasure Island, is above all an honorable and trustworthy young man. He promises in confidence to speak up for Long John Silver if he should ever go to trial, and Silver allows the boy to remain unharmed with the rest of the pirates. When Dr. Livesey boldly enters the pirates' camp to administer first aid to the wounded men, he suggests to Jim that he run away with him afterward. But Jim has made a promise to Silver, who knew he could trust the boy.

"Hawkins, will you give me your word of honor as a young gentleman—for a young gentleman you are, although poor born—your word of honor not to slip your cable?”
     I readily gave the pledge required.

Jim promised Silver not to run, and he intended to remain faithful to his pledge.