Why did he fear such a man as Dr. Livesey?

Dr. Livesey wasn't only a physician..... he was a Magistrate.

The doctor never so much as moved. He spoke to him as before, over his shoulder and in the same tone of voice, rather high, so that all the room might hear, but perfectly calm and steady: "If you do not put that knife this instant in your pocket, I promise, upon my honour, you shall hang at the next assizes."

Then followed a battle of looks between them, but the captain soon knuckled under, put up his weapon, and resumed his seat, grumbling like a beaten dog.

"And now, sir," continued the doctor, "since I now know there's such a fellow in my district, you may count I'll have an eye upon you day and night. I'm not a doctor only; I'm a magistrate; and if I catch a breath of complaint against you, if it's only for a piece of incivility like tonight's, I'll take effectual means to have you hunted down and routed out of this. Let that suffice."