But the effect is mainly in sugary drinks, not diet. In fact, a study found that diet cola decreased kidney stones, although not meaningfully, so it's probably zero effect at all [1]:
> "There was a 23% higher risk of developing kidney stones in the highest category of consumption of sugar-sweetened cola compared with the lowest category"
> "Artificially sweetened sodas were marginally associated with kidney stones, with an inverse relation for colas and a direct relation for noncolas."
So no worries about your Coke Zero giving you kidney stones. The culprit seems to be the fructose, rather, which Coke Zero doesn't contain:
> "Higher consumption of sugar-sweetened soda was associated with a higher incidence of kidney stones, which may be because of the fructose content. Fructose has been shown to increase the urinary excretion of calcium, oxalate, and uric acid, thus increasing the risk of stones."