Larry - It does sound like your bosses are completely incompetent at best. Sexual harassment is a serious accusation, and it just isn't the type of thing that you accuse someone of without having your facts straight. However, my advice would be to let it go on the work front unless you already have some other job lined up. Asking for a written apology might make you feel better and keep your pride intact. However, I can't imagine it will do anything positive for you on the professional front.
Do you think they did it just to pick on you though? Even though the lady was nice and said that she wasn't talking about you how else would they have come to the conclusion that it was you? She must have given such a vague description that they thought it was you instead of the other employee. I guess my point is that something led them to believe that it was you based on what she said. It sounds like a miscommunication more than anything. That is, unless you think your bosses would actively target you. Did you find out who the culprit actually was? If so, do you think it would have been easy for your bosses to confuse the two of you?
Maybe it would be better if you were less formal about it and went to your bosses to ask them how this happened. That would at least let you get their side of the story. If it does turn out that the whole thing was completely unwarranted (i.e. the lady's description of the person doesn't come close to matching you in any way) then maybe you go the route of asking for an apology. If it was completely unwarranted then your relationship with your bosses sounds like it is already in the toilet. Asking for a written apology at that point probably wouldn't be as hurtful to you, because there really isn't any relationship left to get damaged.
For what it is worth, maybe your bosses aren't that down on you. At every company that I worked for you would never get a warning about sexual harassment. You would just get a pink slip, and it would be very tough to find another job. One girl that worked for an old company I was with basically extorted the company after getting sexually harassed. She was a terrible worker, but she got a huge promotion out of it and a very fat bonus that year. That was given to her so she would keep quiet about it. If she had taken it to court she would have taken the company to the cleaners. It was like winning the lottery for her. I wish I had gotten sexually harassed. I would have ended up with a corner office, and I could have always used the attention!
That is just my 2 cents.