I finally get to meet my new supervisor (not my boss, but the team
lead), and I think I messed up. He's a very courteous, but
straightforward, no-nonsense, and extremely large/muscular man. I am
not comfortable around him because he physically resembles a man that
assaulted me years ago (facial features, haircut, etc.). This man is
clearly not the same one that attacked me (the assailant is behind
bars). Nevertheless, I asked HR if I could request a different team
lead, due to the stress it caused me (PTSD diagnosis).
That's good, and that's how you should handle it and HR then should do their best to try to accommodate this request. It may not be instantaneous, and the solution may not be everything you ever dreamed of, but this is a first step towards resolving it.
Due to HR's incompetence, my boss and "Mr Big" found out about my
request, and "Mr Big" has gone from kind to vicious, and is demanding
a public apology (i.e. via e-mail, which he could very well make
public further outside the company) for likening him to "common gutter
trash". If I don't play along, I will be fired and blacklisted with
I don't think that this is what happened, especially when you take into account the following that you've said later, and that you provided no evidence that he has found through HR, but have said:
I may have accidentally said he looks like a man that attacked me to two colleagues.
That's bad, very bad, why would you ever do that? Saying something like that to anyone in the workplace, besides HR, will make this hot gossip spread like wildfire. And it will not be in the original form of "this guy reminds me of the guy who is behind bars.", but something warped into a lot more sinister entity. That is also how it likely reached Mr. Big, who found it out from another employee that someone is spreading such rumors about him, and because of that got, quite justifiably, angry. I certainly would be in this situation.
How can I fix this? Is a public apology just a way to sabotage my
career and fire me without severance pay?
While it is possible to fix it, I honestly would start brushing up your job interviewing skills again as this is a lot of damage to undo. To start fixing it you have two routes, one will be where you meet with Mr. Big and HR in private settings and hash this problem out (don't do a 1:1, make sure to have the HR rep there) and hopefully, when he is briefed on all the context you can move on and agree on bygones.
The other route is to lawyer up. While you may even be successful in this action (although it's debatable as facts are unclear), the rewards in this will be very limited, and you certainly will need to look for a new job, as your future in a company you just sued, as a new employee, will not be bright and great, even if they will be forced to keep you.