The company's (a big American pharmaceutical) HR has investigated their phone, with consent, and was unable to find anything.
They don't need consent. Company laptops, phones, etc. are company property and can be investigated at any time, for any reason, with or without consent or notice. There is no reasonable expectation of privacy in this case.
Case being, is there any point in actually fighting back against the company HR?
This really depends on your jurisdiction and your exact relationship with them. In most jurisdictions, you can sue for being fired for illegal reasons (e.g. discrimination, retaliation, etc.).
Depending on the exact circumstances, you might be able to sue for slander. Obviously, that assumes that a) the accusations were false and b) someone involved actually knew that the accusations were false. You'd have to talk to an attorney about that, though.
If you were, for some reason (company policy, union membership, employee of certain government agencies, etc.), entitled to a hearing prior to being dismissed, you may be able to demand that as well. You could also check to see if there's some kind of appeals process or a way of demanding a hearing within the company itself.
If none of those apply (slander, discrimination, retaliation, right to demand a hearing) and you were an at-will employee, there may or may not be all that much you can do about it. Your best bet could be to check with an attorney to see if they violated any labor laws in your jurisdiction.
This is, of course, assuming that the charges against your friend are, in fact, false. It seems at least mildly strange that the H.R. investigation found nothing, but they fired this individual anyway. Is there any possibility that there's more to the story than you're aware of?