If you're sincerely interested in mending your reputation and relationships in this group, then the first step is to loosen your grip on this premise:
...that I was a racist and made racist comments to him. This is completely fabricated.
Why should you reconsider this? Because the way to resolve an interpersonal conflict is to listen to the other party and attempt to understand their point of view. It's not possible for you to listen to him if you begin with the assertion that what he experienced didn't happen. I don't mean to minimize your response --- it's genuinely very, very hard to put aside your own feelings and perspective when you feel under attack, as I'm sure you do right now. But you asked what your approach should be if you talk to him, and the answer is that it should be one of open-minded listening. In order to do that, you have to allow the possibility that you did say something racist, even if you didn't mean to. Everyone does sometimes, unfortunately.
So let's assume for the moment that you did say something racist to Ed, without realizing it. It might have happened a bunch of times, even. How might that have made him feel? Marginalized? Threatened? Angry? Betrayed? Scared? You might have hurt him badly. In order to repair the damage, you need to put aside your perspective for the moment and apologize for hurting him.
Whether it was intentional or not doesn't matter --- if you hurt someone, you apologize.
As an analogy, imagine if you stepped on Ed's foot accidentally when you got into the elevator, and he winced. For most people, the instant response would be "Oh my goodness, did I step on you? I'm so sorry! How clumsy of me. Is your foot okay?" Imagine how outlandishly rude it would be to respond with "Why are you wincing? I absolutely did not step on you. I'm not the kind of person who steps on people's feet. How dare you accuse me of such a thing. You owe me an apology." And yet, we well-meaning white people often have a very hard time seeing the effects of unintentionally racist comments as harm because we can't see past our own pain at thinking (or having others think) we might be racist.
"But his accusation was completely fabricated!" you might be thinking. "I never said anything to him that could possibly be construed as racist." Well, there are two possibilities:
You did indeed say something that he interpreted in a way different from what you meant (e.g. you said something you didn't realize had racist connotations), so while he genuinely did experience you saying something he perceived to be racist you have no memory of having done that.
He has some kind of completely unfounded personal vendetta against you and is maliciously spreading fabricated stories that he knows to be false in order to hurt you.
In my experience, #1 is much more likely. So start with that assumption and try to resolve the conflict from there.
How can I recover from this in my workplace? Here are the steps I recommend
- Read this excellent article in its entirety: Getting better at getting called out It includes tips for helping you to take Ed's perspective here, as well as very concrete dos and don'ts for talking about this with him.
- Take some time, alone, to prepare yourself. This will be a very hard conversation --- don't underestimate the power of preparation. Think carefully about what you want to say (and what you want to NOT say), and how to say it. Write down talking notes for yourself, and practice saying them.
- Ask Ed if you and he can have a talk about this. Here's an example of what you might say to ask him:
Hi Ed, I had a conversation recently with HR and they told me I had said something problematic to you a while ago, something that sounded racist. I was really shocked and dismayed to hear that; I hope you'll accept my apology. It was certainly never my intention to contribute to a racist environment here at work, and I sincerely apologize for my actions. If you're willing, I'd love to talk with you more about this so I can understand exactly what it was I said that caused that reaction, so I can do better in the future. Either way, thank you for speaking up about it.
A couple things about this example script: It starts with the assumption that Ed's experiences were real; he experienced you saying something that sounded racist. (In our elevator analogy, we're acknowledging that he got his foot stepped on.) Because you're starting from a position of validating his perspective, a heartfelt apology for having unintentionally hurt him is the natural response. It also puts you both on the same team by thanking him for speaking up --- it makes resolving this and preventing this kind of thing in the future your shared goal. In that spirit, it asks him to help you learn more about what you did so that you can avoid making that mistake again. Note that it's not his responsibility to teach you, though, and he has every right to say no. If he does say no, accept that gracefully and try to do some honest introspection to see if you can get to the bottom of the matter on your own, or with a trusted friend.
If Ed takes you up on your offer to talk about this issue, concentrate on listening with an open mind. Resist the urge to tell him your perspective, to correct his impressions, to explain yourself. This is NOT the time to explain why you're not racist and his impressions were wrong. Your goal is to understand what his experience was. Keep in mind that this is a hard conversation for him, too, and he's doing you a really big favor by taking the time and emotional energy to engage with you on this.
Work on humility on this front going forward. It's much better to apologize right away when something like this happens, rather than it festering over months. Dismissing it (saying it didn't happen) doesn't make it go away. As soon as you think you've said something that someone was offended by, try to correct it. If you and Ed do have a talk (#4 above), maybe end it with something like
I really appreciate you taking the time to talk with me about this. I know this isn't an easy this conversation, and I'm really grateful [warm smile]. I'm going to try to be more aware of what I say going forward. If I do step out of bounds accidentally, I hope you'll feel comfortable calling me on it.
And hopefully he will. Immediately, and to your face (rather than behind your back to other coworkers). If he --- or anyone else --- does call you out in the future, be very careful about managing your reaction. Re-read that article on getting called out every once in a while to keep those strategies fresh.
Good luck! :)
NB: Since you posted this on IPS and not on Workplace, I am responding to this in terms of how you can mend your personal relationships at work (rather than how you can protect your job, maintain professionalism, avoid getting into trouble with HR, etc.). If you want help addressing the workplace side of this problem --- which admittedly is not what you asked --- you can ask another question there. I just wanted to mention that this is a problem with multiple facets, and my answer only deals with the IPS side of it (as did your question).