As I understand the situation, a close friend told you that a mutual acquaintance (a coworker?) accused you to said friend of having emailed him an objectionable email? That as evidence this acquaintance provided your friend with the email header which included the (static) IP of your office or home computer?
Under other circumstances, I would say that anyone who accuses me of emailing an objectionable message is a bit of a fool, because as others already pointed out, it is all too easy for spammers (and other bad guys) to forge email headers. But in these cases, I would not expect to be told that the recipient is acquainted with me.
Is the person who recieved the objectionable email in fact in your on-line address book? Or that of your mutual friend? It could be that some virus on the machine of your mutual friend might have been used to mail objectionable messages allegedly "from" some persons in his on-line address book to other persons in his address book. However if, as I guess, all three of you work in the same building, the most likely explanation might be a prank by a coworker, possibly someone who had access to your computer while you were out of the office for a few minutes?
I know you said "I am the only one with access to my computer, no one else could have possibly used it", but "no one else could possibly"... famous last words, eh?
What can I do to hopefully prove my innocence?
Well, you really can't prove
your innocence. But once the injured party has had a chance to calm down and investigate alternative explanations of how whatever occured might have happened, and why, he might recognize that you are (perhaps) an unlikely person to send out the kind of objectionable email he says he received.