A single action revolver like the model he had won't allow you to lower the hammer back down unless you pull the trigger. The trigger has to be held down the whole way down or it will catch on the half cock notch and then jam up. There's no way it happened unless he was pulling the trigger.
Now, given the information we know, I don't know that I would find him at fault for not checking the gun. The apparent standard procedure with guns is ridiculous from the perspective of someone who has been around guns my entire life, but if it is standard procedure I think that is more of an indictment on Hollywood than Alec Baldwin. In my opinion, if you are going to be handling firearms on a movie set, you should be required to be familiar with them and safe handling of them just like with any other job that involves potentially dangerous tasks, but that isn't how it is. The practicing your draw with people in the line of fire is really the big issue. They should be a huge no no according to normal firearms safety. On the one hand, it is pretty much required at times due to the job description. But on the other hand, Hollywood doesn't follow normal safe handling procedures and it is accepted, so maybe he shouldn't be culpable there as well.
I think what this will come down to is how much power did he have about hiring decisions with regards to the armorer. Their were apparently a lot of people who weren't happy with the way she was handling things, so much so that they almost walked off the job. If he had the power to make decisions regarding her employment and ignored those concerns and still did what he did without making sure that everything was safe, then he may have some culpability.