My DH did not cry immediately after birth, nor did I (which surprised me, but I was in such shock I was just trying to take everything in.
DH did cry later that evening though, when it was finally just the three of us for the first time. He sat next to me on the bed and started to pray over our new family. As soon as he started to speak the tears began to flow, which then triggered the ones I hadn't shed all day. It was such a sweet, tender moment - one I will cherish forever.