Am honoured you linked me here, thanks.
pKa of the important proton in uric acid is 5.8. Below that pH the molecule is uric acid and is not charged. At physiological pH (7-range) it mostly exists as the urate ion (single negative charge). The urate ion is more soluble than the uncharged uric acid form. So at lower pH, you get more uric acid, which is less soluble, and once solubility threshold is reached, precipitation of uric acid stones ensues.
From UpToDate:
"Uric acid solubility in the urine falls from approximately 200 mg/dL (1.2 mmol/L) at a urine pH of 7 (a setting in which 95 percent of uric acid is present as the more soluble urate anion) to 15 mg/dL (0.09 mmol/L) at a urine pH of 5 (a setting in which most of the uric acid is the less soluble, undissociated acid) [
1,10].
The importance of a low urine pH was illustrated by the biochemical characterization of 341 uric acid stone formers from four studies. The mean urine pH in patients with uric acid nephrolithiasis was 5.4, and virtually all patients had a urine pH less than 6 [
4-7]. "