Workplace question: How would you deal with a manager/superior who is apparently constantly looking for the smallest mistakes in your work and acting highly hostile?

I am part of a managerial team at a small business, and I've worked there for 3 years. I started out at the bottom, "floor staff" so to speak, and through hard work and experience, worked my way up to manager. For the entire time I was working there, I had a fantastic relationship with everyone, and I considered everyone to be a good friend.

However, recently, one of the people who mentored and trained me for my managerial duties has changed entirely in his attitude towards me. This is a single example of many that characterize our recent interactions:

I was leaving at the end of my shift, 10 minutes late so as to finish up something I started, I noticed something was wrong with some equipment and that it might cause an issue in the near future, but it was an issue that could not be rectified at that exact moment. He was relieving me, so as I was briefing him, I said, "by the way, just now I noticed an error message on X machine that might need to be fixed before running it again."

I got an email later that day saying, "Mumbojumbo, the error message you told me about was timestamped to X:XX AM, meaning that it was there for the duration of your shift. This is unacceptable and irresponsible, and you need to do a much better job with your duties." This email carried on for several paragraphs.

He CC'd the COO of the company and head manager of our location on the email.

I emailed only the coworker in question, saying, "Are you sure that the error message was from that time? I double checked the machine when I got in and it wasn't showing any errors. Are you sure that the clock on the machine is calibrated correctly?"

He emailed me back saying, "It would be good to know if the clock is wrong."

He worked twice in the following days and never mentioned the email to me. I checked the clock and the clock on the machine was set wrong, in such a manner that proves the error was about 30 minutes old when I discovered it. I found a note by him on the machine log saying, "Machine displayed error since X:XX AM, fixed at X:XXPM." I corrected the first time to reflect when the error actually occurred.

I got a furious set of text messages from him about 2 hours later, while at home, basically saying, "NEVER REPLACE SOMEONE ELSE'S NOTE, I can't believe you thought that was okay."

I don't know how to approach the situation. I've already discussed his attitude with him, kindly requesting that he continue to tell me when I've made a mistake, but just to do it in an unaccusatory tone, so as to not breed hostility. He said he'd "work on it" but it's become apparent that he isn't. I'm worried that, since he is above me, and in constant communication with all of my superiors, he is affecting the way I am viewed within the company in a negative way. His attitude towards me is making it very difficult to have a constructive work environment, and it is breeding hostility. I am the only person he seems to take issue with, he is very outwardly kind to all others in the workplace. I have received nothing but positive reinforcement and thanks for my work from my other managers, along with the occasional polite correction, but from this one guy it's nothing but vitriol and hostility. What would you do in my situation?