Heathcliff embarrasses Hareton most by continuing to raise him in the fashion of his revenge. He is determined to treat Hareton, the son of Hindley, worse then he was treated by Hindley. Hareton was never taught to read or write, never rebuked for any of his bad manners, never corrected when swearing, and never taught refined manners. This led to embarrassing moments with Cathy and Linton later. Both of these young people made fun of Hareton and called him names. It came to a point with Cathy that Hareton refused to have anything to do with her, which Cathy later regretted.
Twice, Heathcliff embarrasses Hareton verbally. When Linton arrives, Heathcliff is disappointed in what he sees. He wants Linton to marry Cathy and inherit the Grange since he already has Wuthering Heights, land that once belonged to Hareton. When Nell asks Heathcliff to treat Linton with kindness, he replies,
"And to begin my kindness, Joseph! bring the lad some breakfast. Hareton, you infernal calf, begone to your work!" (pg 201)
The second time he embarrasses Hareton is when he asks Hareton to show Miss Cathy around the property. He says in front of everyone,
"You'll be the favourite among us, Hareton! She says you are --- what was it? Well, something very flattering. Here, you go with her around the farm. And behave like a gentleman, mind! Don't use any bad words; and don't stare, when the young lady is not looking at you, and be ready to hide your face when she is; and, when you speak, say your words slowly, and keep your hands out of your pockets. Be off, and entertain her as nicely as you can." (pg 210)
Afterward Heathcliff tells Nelly ,
"I've tied his tongue.....He'll not venture a single syllable, all the time" (pg 211)
Heathcliff has also ordered Hareton to obey Linton. (pg 201)
The page numbers I have given are for my edition of the book, but you should find them somewhere in the vicinity of those pages.