Well, if you look at the sequence of events, I think it's pretty clear that the "fate" he was referring to was Rita's death.
The thing is, Dexter's whole approach to being a serial killer is that there are certain people who deserve to die. That's what the Code is all about (well, that and not getting caught). He bases everything he does on the idea that people are morally accountable for their actions (he excuses his own actions with the rationalization that he only kills "monsters").
If he takes Arthur's "fate" approach, practically everything that makes him what he is goes out the window. He doesn't follow the Code anymore - just kills whomever he's "fated" to kill (like Arthur did).
I could see him turning away from the path to humanity, deciding that being what he is is "fated." But I don't think it's possible for him to go the way Arthur did with it.
As for the kids, I predict that he won't keep them. He MIGHT keep Harrison, but look for Rita's kids to go to their grandparents or another relative. Not only would it be impossible for him to do what he needs to do and take care of three kids by himself, but I don't think he'll want to now. I think he's decided that family ties are liabilities.