I deal with a lot of politics in my family. I recently got into a pretty heated discussion with my Dad, but we both made it out with our relationship intact. Here are some tips to help you get through this.
First off, your both adults. You are both entitled to your own opinions, even if that means they are on opposite ends of the spectrum. A book I read a long time ago helps deal with this, its called: How to win friends and influence people by Dale Carnegie. One of the topics covered in the book is how to win people over to your way of thinking.
You need to find his factually incorrect posts and correct them. Confront him about them and present your case. Don't yell, try not to get angry. Try to tell him in a mild pleasant demeanor. Be persuasive and help him see your side of the picture. His response will be to dig in his heels and refuse to accept it, but you need to avoid any conflict.
The only way to get the best of an argument is to avoid it. Whenever you argue with someone, no matter if you win or lose the argument, you still lose. Your Dad will either feel humiliated or strengthened and will only seek to bolster his own position. Try to avoid arguments whenever you can. An honest discussion is very different from an argument. Arguments usually involve yelling and "your wrong, I'm right" attitudes. The objective is to help him see your side of the picture.
Show respect for your Dads opinions. Never say "You're wrong." Never tell your Dad he is flat out wrong. It will only serve to offend him and insult his pride. No one likes to be humiliated. Start the conversation with something like: "Hey Dad, I saw your post about [FB Post] and I don't think thats right. Can you show me some sources to prove this? Because what I've got right here says [FB Post correction]."
Try honestly to see things from your Dads point of view. Other people may often be wrong, but you cannot condemn them. You must seek to understand them. Success in dealing with your Dad requires a sympathetic grasp of his viewpoint. You can't argue against something you don't understand. What caused him to think like this? Did he have any influential people is his life?
And finally, be willing to accept the fact that you might actually be wrong. Your Dad has been alive longer than you and he knows his stuff. Unless you have evidence showing otherwise, you might just be getting annoyed at him because you don't like the way he thinks. Most people aren't aware of the opinions of people around them. Your Dad might not know that you don't like the kinds of things that he posts on FB. Unless you verbally announce to him that his posts are hurting your feelings, then he isn't aware of it. He can't read minds.
You are both human. You have differing opinions and they are causing conflict. It happens more often than not. Just work through this. The relationship won't rebuild itself. You have to put in what you want to get out. If you want love, respect, and recognition, you have to put in love, respect, and recognition.
Alternatively, you could just unfriend him, get rid of FaceBook, and never talk to him again. I wouldn't recommend this though.