Well firstly, you did your best with the knowledge you had at the time. Anyone who appears not to understand that is either very inexperienced or just pretending. Anyone with experience knows that these things happen.
So you do not need to over apologise, you didn't have a crystal ball, and you had to make a call.
Secondly, in these situations I the main thing to think about is what does the other person want? Mostly people want to feel heard - that usually takes a lot of heat out of anger - he probably understands that you made a reasonable call at the time, but the board is yelling at him so he's expressing his own stress.
You can make him feel heard, and empathise with without over apologising or owning fault that does not exist.
Also, make sure you have ideas on the strategy to deal with this problem, if you are asked (don't volunteer them unless you get the vibe you should).
Thirdly, once you've let him have a rant, empathised and offered your opinion on strategy if asked, let silence fall. The chances are you won't be fired or made to walk naked in penance through the office. Shit will get dealt with and life goes on.
Finally, it sounds like you have a great team - despite them supporting you in this crisis, they still want to have confidence in you as there leader, so don't apologise there either, just listen, appreciate and empathise if needs be.
Finally, finally, in the unlikely event your head is needed for political reasons, don't stick your head in the sand about that - see your union/a lawyer faster than you can say settlement, and make sure you get a top notch pay off. No one wants a disgruntled leaver, they'd far rather buy an amicable departure.