I've been in this situation a couple of times in the last 2 years and it is so hard. I found that I had to make the decision to try and take all the pressure off him for a short time, so he could truly rest and recover. I know that must be really hard with a baby but it was the only thing I could do to help him get through it.
It's good to let him sleep if he needs to as depression is exhausting, but I mixed that with nagging DH to get out of the house for a run or a walk sometimes so he didn't just fester. I let him hang out upstairs on his own if he needed to, and he would come down and join in with the family for as long as he could manage.
One approach he read which he found helpful, was that you find something to occupy your brain, without having to actually think too hard, DH did jigsaws, sudoku, played various word and number puzzles on his phone. It stopped him dwelling on stuff as much.
I found I had to let him be selfish for a little while, and I took on the responsibility for pretty much everything, but I think he recovered quicker that way than if he had continued to push himself. It sucked and I cried lots and was exhausted, it's horrible for both of you.
Make sure he gets to the GP and considers taking ADs, counselling is great if you can get it (we went private).
Really hope things get better for you both of you soon