I was in a meeting with our CEO and he was taking feedback for my supervisor. One thing lead to another and I mentioned something that I believe I shouldn't have. He wanted to fire my supervisor and replace him. That didn't happen however my supervisor knows about our conversation and I believe there's a chance he might have seen some text we exchanged between us. What should I do?
You stabbed someone in the back and now you're worried they may retaliate. Your best bet would probably be to just pretend it didn't happen and avoid a potential confrontation. Unless you were lying I see no need to apologise. Quite possibly the manager is more scared of you complaining again then you're worried about them. The chances are that they got a good telling off from the CEO and will be tiptoeing around for a while.
This person will never trust you again in all likelihood, be careful and professional around them.
Firstly, why shouldn't you have mentioned it? If the feedback was negative but fair then I wouldn't be too concerned.
Assuming you had a one on one discussion with the CEO that was for the purpose of collecting feedback then I wouldn't expect that the CEO reflected the comments directly to your supervisor as being from anybody specific. He likely knows that somebody(s) in his team have made a complaint, but it's unlikely that he knows the who or specific details about it. I would carry on as per normal but expect that he's likely going to be looking to redeem himself, potentially at the expense of his team so keep all conduct as professional as possible.
If however the case is that you were unprofessional in your complaint and you feel it's not something that should have been mentioned then I would defer to @Kilisi's advice and potentially consider retracting the comments with the CEO (depending on the seriousness of them).
Your supervisor did something that is a fireable offence and you think you shouldn't have told anyone when you were in a meeting for that purpose and now you're wondering how to apologize to your supervisor?
Long ago, one of my children cheated on a quiz at school and came and told me. I told her to tell the teacher what she had done, but she didn't want to because then the other girl would get in trouble. (It was a "pass your papers to the left and the other can mark them" quiz: by prior arrangement she left some blank and the other girl filled in correct answers for her as the teacher called them out, thus confessing would uniquely identify the other culprit.) I told her, and I'm telling you:
If she gets in trouble, it's because of what she did, not what you said.
You were asked about your supervisor. You chose not to keep a secret for your supervisor. Why? Because you have a lot of honor and integrity and put the company first? If so, be proud of that. Because you were scared of the CEO and he stared at you with piercing CEO eyes and scared the secret out of you? If so, own that too - perhaps you have a chance to learn something here. Because you hadn't thought about the consequences of telling anyone whatever you told? Well own that too and this is definitely a learning experience.