Ah, I thought you were wondering if the page numbers get updated correctly.
No: you cannot have automatic renumbering without having InDesign open and update each document. The only way you could do that is by doing the exact opposite: set the page number manually (but then you cannot move individual documents around anymore).
The “starting page number” does not 'live' in the Book document, they are stored inside each separate doc, as you can see if you open one of them without opening the Book file first: the starting page number will still be correct in relation to the entire book. The only function of the Book file in this is changing it when necessary.
That said: if you changed one document but did nothing that ought to affect the other documents, you only have to update that one. The other files may get 'touched' by InDesign (if only to check if the page numbers are still correct) but if your page count did not change, they won't actually be changed.