You write that you're upset that your brother is too stubborn to apologize, but what do you think he should be apologizing for?
Giving you only a day's notice of his visit? I'll concede that it was inconsiderate. Unless his plans changed on short notice, he should have let you know sooner.
Staying only three days? Three days is an awfully long time to spend with someone who's radiating "hovering disapproval," dislikes your wife, gives unwanted advice, and tries to overstep boundaries you set for dealing with your kid.
"Giving attitude" when you asked how long he would stay? Given your hurt feelings about the length of time they spent with you and other family, is it at all possible that you were guilt-tripping him?
Spending a lot of time asleep/not interacting with your family? it sounds like the atmosphere was already unpleasant during the visit. What would you have preferred - your brother having it out with your dad about everything? Would direct confrontation about your dad's poorly-concealed anger have resulted in a screaming match? Your brother's family being thrown out? Silent treatment? Would you be ok with this happening in front of your nephew?
Not visiting relatives X, Y, Z while in town? That's really your brother's call, and shouldn't involve you. Are they close? Are they critical of his wife? Do they have anger problems? Would they pressure, criticize or guilt-trip him on your dad's behalf? Was your brother already tired from traveling with a kid?
Posting photos of your nephew to a shared album, showing the fun things he's doing without you? It sounds like you and your dad are really reaching to find something to be offended by. I'm guessing that, if they hadn't posted photos, you would be angry that they didn't care enough about you to keep you in the loop about nephew's adventures.
Refusing to give your parents unsupervised access to their kid. This is their decision. As you describe him, your dad is angry and hypercritical, and they can't count on him to keep his temper under control if he has their son alone. They can't count on your enabling mom to protect him from your dad, either. You're also upset that they won't let your mom take their son to visit the neighbors. The problem is that being inconsistent with this boundary makes it more difficult to enforce when it counts. If your mom is allowed to take their kid somewhere alone, why can't your dad? If your mom can take their son to visit the neighbors, why can't she have him over when your dad's at work? You mention that you find this overprotective, but ultimately this is your brother's call, and you have to accept it.
And if the fault is on both sides, why were you only nagging your brother to apologize and reconcile? Is it less scary to confront him than your dad? Do you think he has some obligation to be the bigger person, while your dad is just too set in his ways?
I think you're coming from a kind and concerned place, but you really need to examine your assumptions before trying to repair this rift. And as for the letter from your dad - no response is necessary. Ignore it, the way you would ignore anyone else embarrassing himself in public.