I have a slightly different take on this. I think little kids find it slightly terrifying that they have the power to hurt their parents. They need to know that their parents are strong enough to still be calm and in control and emotionally okay, even if they (the kid) flails against them.
This struck me from the original post: "After his massive tantrum this morning he started crying and saying that he was afraid no one in the family would like him anymore because he is so mean." To me that sounds like a kid who is already freaking out about his power to hurt other people.
Like others, I wouldn't let "I hate you" slide. But I would say something like, "Wow, it sounds like you're really having strong feelings." Later, in a calmer moment, I would have a discussion about the behavior and why it's not okay, what some alternatives are for dealing with feelings, and why an apology is necessary when you do lose it.
In other words, I wouldn't ever put it in terms of "You hurt me, and things are not going to be okay unless you do something to un-hurt me." That's giving a child too much power in the parent-child relationship, power that they are not equipped to handle.
