As I read it: (Conjecture, really)...
We've already seen that Alice and Sedna have a thing going where Sedna's abilities aren't as great as Alice's, nor does she feel a personal duty to a community and human welfare in the same way that Alice does. We've already conjectured that Alice is a "more advanced" or later version of whatever advanced weapons program produced them both. Church is a "More advanced" model than Alice and Sedna. He was developed later, when both the knowledge about how to make somebody that strong and about how to keep somebody that strong under their control was more advanced than it was when Alice and Sedna were produced.
So when it was over - when the old world had gone to shit and there was really and truly no more point in fighting - Church was the one who couldn't stand down. He had orders from some recognized authority and could not overcome his compulsion to obey them. He was stronger, faster, more durable, and as ruthless or more ruthless than anyone else standing, and could not stop fighting. Alice felt duty to human welfare and so on, but she was free to reinterpret her duty and make choices about how to carry it out, and because Church didn't have that freedom, he had to be taken down. And taken down he was, with Alice to wield a hammer and an orbital laser strike to make a handy pool of lava.
Now that Church is above ground again, we see him under the exact same compulsion he could never overcome, to obey orders from some recognized authority. I don't think loyalty or gratitude or valuing that entity as a person or anything like that comes into play here. Church is literally stuck on a compulsion, as in an obsessive-compulsive disorder, that he can't overcome. Pate is some kind of recognized authority figure, whether he sees Pate as a worthless worm of a man or not, whether Pate ever did anything for him or not, and when a recognized authority figure gives an order, Church obeys it.
And now Alice is laughing at his predicament. His is greater strength than hers, greater speed, greater endurance and durability, but this one crucial weakness - the very same weakness that forced her to put him down fifty centuries ago (possibly even the same type of threat that forced him to stand still for it) - prevents him from taking his revenge on her. That's some high-grade irony there. Her sanity hasn't broken; there's an element of nervous laughter there as she experiences relief instead of death, of course, but this is to me clearly a laugh at a situation which is funny as hell if you have a very very dark sense of humor.