How do I settle a misunderstanding with someone who thinks that it doesn't matter what my intentions are, it's how I said it?

So who "puts words into your mouth"? Are you a puppet? Is there a ventriloquist in the house?  It may also be that the person who hears these words interprets them differently than you intended, but it was you, who said them.  You appear to have communication difficulties in any case, see a counselor, who can teach you both better communication and problem solving skills. Without these, nothing else will be resolved. 

If you have a baseball bat, now would be a great time to USE it!

That isn't an argument you can win.  It doesn't make any difference what your intentions were when you spoke the words, what is important is the perspective of the listener.  What you can do is learn from the argument to not use any words that might be misunderstood with that person in the future.  You may not realize it, but you treat different people differently because you have learned what is and isn't acceptable with each person.

Unfortunately, it often doesn't matter what a person's intentions are because people react to what they receive and what people receive can be very different than what we think we are giving. Sometimes I think people are receiving offense because they are seeking offense. That being offended rewards them in some way we can not perceive. Apologize for the offense and drop it. You can not control their choices or their perceptions of reality. It's likely you simply don't get to control this. They are offended. Don't explain. Apologize and drop it. They are making their own choice. 

Sounds like the other person talks a different language to you.  I would stop apologising and leave matters as is.  Agree to disagree.  Don't try defending your actions/speech.  He is determined to NOT  to understand because he wants to blame you.

"...all the word being put in my mouth aren't mine."

Huh? 

I have no idea if you're trying to excuse what you said (you conveniently left out what that was) Or maybe your acquaintance no longer wants to be your friend (for whatever reason) and is trying to break off the relationship. If that's the case, walk away. You can't force someone to be your friend.