You have apparently not been in the workforce too terribly long or you'd have a little more perspective about the relative importance of this "workplace drama" issue. It's not such a big deal in the course of a work life, honestly. You'll not be losing sleep about this years from now. It's not a high point in your life. I've seen people fired for things like this (though it will not probably not considered a misconduct offense to the extent that you'll be denied unemployment, though we can't really tell). You are in an "at will" state. You could if they wanted to be fired because they didn't like it that you said she has red hair, when she does have red hair. But I suspect it will blow over. And even if you were to be fired, hey, it will happen a time or two. It is NOT the end of the world if you screw up at work, or even if you get fired from a job.
What to do now? You made a stupid mistake. Oops. Tell HR that's what it was. Apologize, briefly and professionally. Something like: I'm sorry. I had some personal knowledge about the situation that led me to say some things I shouldn't have. I overshared. I realize it was inappropriate. I will not do it again. (Then you stop talking.)
They may want to give you a write up. Don't make a big deal of it. They may just rebuke you. Don't make a big deal of it. Go back to work. Be quiet, professional, and MOVE ON. Say no more about the issue. DO NOT try to go to the person and apologize. DO NOT go back to the people who passed this on and discuss it further. DO NOT be the one who ever brings this up with HR or any of your co workers. Move on, you are an army of one when it comes to your work career. Be a professional, be proactive, and don't make this mistake again.