You note that the head of HR has "basically" accused the employee of lying, but "basically" accusing someone is rather different from "clearly", "certainly", or "actually" accusing them, and it leaves a lot of wiggle room in what this accusation could actually comprehend.
From my perspective, the best course of action in this circumstance will depend 1) upon just how "basically" or "actually" the accusation was, and 2) your own tolerance for risk and exposure. I'll outline what I think would be the most sensible responses for different kinds of situations below.
"Accusations" can take many different levels, from (1)"implying that someone was wrong (and therefore not truthful)" (ie; "it seems that the deliverable was not completed on time, as your communication implied/seemed to claim you had done"), to (2) "directly questioning someone's memory (or honesty)" (ie; "are you sure that you finished that as you have stated? I have reports that it remains incomplete"), all the way up to (3) "direct accusation" (ie; "Your claim that this work was finished is not true. It is not finished. Why have you made a false claim?")
If you have evidence that an accusation is unfounded, the type of response that will be merited obviously changes with the type of accusation. (1) and (2) maintain a friendly facade which at least appears to allow for "misunderstandings" and "clearing things up". If evidence contradicting the accusation is to be provided, then it should also maintain this friendly facade. ie; "Oh, actually I can confirm that this was finished on time! I have {this evidence}. Is it possible that there was a miscommunication, or one of the metrics wasn't updated? Let's check this out!" Even if the accusation was sinister at its core, these weak accusations have the benefit of letting the accuser disclaim them with equal ease, and just "clearing up the misunderstanding" may be all that can be done. For an employee with no skin-in-the-game (or even the accused themselves), other than this kind of "friendly" rebuttal, there isn't likely to be much more that can be done. And, after all, retaining that "friendly" professionalism will reflect better anyways.
Only a more serious accusation could even possibly occasion a more serious rebuttal, with more forceful insistence on the rectitude of your own evidence. And I have serious reservations that there will almost ever be a business case where an average employee will be justified in being so forceful. But in whichever case, your assurances had better be no more sure than your evidence is; don't over sell it or claim to know for sure things that you only believe. And don't become a partisan; as an average employee you are a witness, not a member either of the defence or the prosecution. (Unless you have a very high tolerance for risk, a very high moral code, or don't care especially about this job)
In this case you make it sound like the accusation might be (1) or (2) but certainly not (3), so keep that in mind.
The next question appears to be: how appropriate would it be for you to enter the fray, at all, regardless of the level of the accusation? Companies are, generally, not medieval king's courts, where the average person can come in a present a petition whenever they like. It will be better for your position, and lend more appearance of reliability and lack of bias, if you have some kind of "standing" to enter the fray. After all, there seems to be a natural distrust of people who barge in, uninvited, with a bone to pick.
In your case, you say that this exchange took place in an email chain, including the CEO, the head of HR, the accused employee, and (apparently) yourself. This would seem to give you a reasonable, low risk, and well justified opportunity to respond to the accusation, regardless of whether it was a light contradiction or a heavy, crucible-like accusation. After all, you are a member of the organization (ie; you have an interest in its workings, and in "doing what is right" for said organization). You're part of a communication, and you have information relevant. If for no other reason, it makes sense for you to provide it, even if you won't make the call yourself. After all, your hands don't make the decision what to touch or what not to, but they will still tell you that the stove top is really, really hot. (Luckily, unlike hands, employees are at least theoretically capable of abandoning organizations that would "put them to the flame").
If, however, you don't have "standing" at present, then I would recommend finding another way to present your information (if you feel it necessary).
The accusation is currently known to you, as a fact. It is not an office rumor; you saw the emails. Presumably, you were allowed to see those emails. Intended to, even. If not, then it should be treated as an office rumor until you learn of it through more proper channels; an easy way would be to ask someone who was directly involved about it (as "something you heard"). But whether it is a known fact, or a "rumor", if you aren't in a position to participate in the email chain (the "place where it happened"), then you would have to find an alternative approach.
First would be to approach one of the principals; either the accused employee, or the accusing HR head. If you could at least reasonably believe in their good-faith, and goodwill, then you could offer one or the other of them your information. "Hey, HR head, I saw that there was some information of X, but I have this information that says Y. Could there have been some mistake?" or "Hey, employee, it seemed that HR head thought that you had done X, but I have some information that supports Y. If you needed any help clearing it up, let me know!"
Of course, if the interaction is already contentious, or you don't trust one or the other of the parties, then this amounts to taking sides in a brawl. And that exposes you to risk. The alternative is to provide the information to a third party; someone influential (in this case like top management), who is also apparently disinterested in either of the parties' cases, and who (hopefully) you trust. ie; "Hey, mr CEO, I saw that there was some conflict between HR head and employee. I don't know for sure, but I think there might be a miscommunication somewhere, because I have information Y. I wanted to provide it to you, because I knew you would be able to judge it. I didn't want to get caught up in anything or contribute to any conflict."
These appear to be the ways in which one could put forth their information without appearing partisan or exposing themselves to undo risk. Of course, if the accusations get more serious, then participation will consequently entail more risk. And all risk is, obviously, dangerous. Which is why the final question is; how much are you willing to stick your neck out, and for what? Are you interested in protecting the employee? Are you interested in punishing the HR head? Are you interested in helping your company? Are you interested in morality, and seeing right done? Or, perhaps, are you just interested in keeping your head down, making your paycheck, and avoiding getting burnt by someone else's fire? Take your pick. But they all have risks; even trying to stay neutral.
