Obviously we weren't there and didn't see what happened, but from what you've said, your dh had three hours sleep after a night shift and has recently started taking antidepressants, and was trying to manage a crying baby. He made a serious error of judgement. However, those are pretty challenging conditions and if anyone is going to make an error of judgement, those are the kind of circumstances where it could happen so easily.
That's worth re-reading; OP.
What to do now is a tough call. I think I'd be tempted to call my mum (if I had one, and she was supportive, which it sounds like yours is) and tell her that DH is really, really struggling and doesn't feel safe around the children. I'd ask if she would help.
I'd then make sure that DH had no unsupervised contact with the babies. Not no contact at all; I don't think, because that will affect bonds, but I'd make sure I was properly concentrating on him and the babies when they are together. Like you would with a small child; I guess.
And I would take DH to his GP. Take a baby or both; if you need to, but you need to be there if he'll let you to make sure he's being honest. If it's impossible for you to be there; I'd ask him to give the doctor permission to talk to you, so you can let them know how bad things are - but again; it would be more effective to be there together. It's easy for things to be forgotten otherwise.
Can you afford for DH to be off sick for a while and try and recover? It sounds like his health is deteriorating quickly and night shifts aren't likely to be helping with that. If you can; I'd be looking at potentially getting him signed off and involved in some more intensive treatment. Does he get an medical cover through work? Could you afford a private appointment?
You have to protect your babies first and foremost; absolutely, and you need to be certain they are safe with whoever you leave them with. Unfortunately, he can't be trusted to keep them safe right now, he's not quite capable. But that means that he's likely to be very poorly himself; so I'd try and get him help quickly too. He sounds like he's balancing on a knife edge. Depression can be an awful thing.
I'm sorry that this has happened. Could friend come round to you later? Different type of evening to what you'd planned; but it might be the best you can do right now.