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@gbj_tester
Oddly enough, that's not even the right answer in theory.
If you take a theoretical tyre, with no casing thickness, it ends up looking like a circle with a diameter of the cross section size tangential to both rim bead hooks. A bigger circle will have it's centre higher above the bead hooks if you keep using the same rim*. Furthermore, the inner surface is the circular section on a practical tyre, so the tread band thickness is added on top, and that's probably going to be thicker on his replacement tyre because he's looking for something more robust.
In short, if you go from a 23c race tyre to a 28c touring tyre, you might add 8-9mm to the rolling radius, assuming both tyres are true sizes. As usual, all bets are off with commercial tyres, because few of them are the size written on the side, although I suppose they all are if you select exactly the right rim width to make them so.
*for the same reason, using a rim with a wider gap between the bead hooks will decrease the rolling radius for a particular tyre.