If he is the cause of the disintegrating it'd be because of his power or how long he's used the body. Either way, I'm surprised he's still using it, so I'm going to take the crack theory route for now until proven otherwise because... I just like it XP
Unless Yuri died before giving birth and then it became time to cut her open.
...that sounds pretty bad...I forgot the official name of it... and I was brought to this world in that way
Well if he had enough time to kill Yuri, he might as well should've aimed at Fujimoto (assuming he was the only one present. I can see Mephisto staying a safe distance away and making Shiro fetch the kids and come back - maybe that's just me) and attempted to kill him.
...unless of course Satan didn't find out until after the fact. But still - wouldn't Lucifer be able to get the kids back if that were the case? Maybe Satan thought at first he'd just possess Shiro, get it over with, give the kids to Lucifer and away we go. But then Satan found it wasn't as simple as that, and Lucifer just went ahead on worked on his Illuminati plans?