Soak the pool balls in a solution of white vinegar & water for several hours, even overnite. The vinegar reduces the water's normally higher ph value and thus lowers it to an acidic level......well below 5.0 at a 50% diluted strength of 1/2 & 1/2 mix. You can even use a stronger mix ratio and soak for a shorter period. The soaking allows the new deposited film (solid matter) to relinquish its attachment to the pool ball's surface area. Every solution contains dissolved solids which consists of various, different compounds. Under different conditions, the substances precipitate out of solution (sometimes due to exposure to high heat, cold, age of the product solution, sunlight, etc.). Even under highly diluted solution strengths (>10-15ppm), the dissolved solids in some products will precipitate and bond to hot surface areas the solution comes into contact with, ex., boiler tubing. These tubes are very costly to replace and typically when deposit formation accumulates, the boiler is descaled using diluted acid solutions.
You can use this same principle (vinegar & water) to remove calcium carbonate water spots from glass windows, windshields, mirrors, & shower doors. Use an old newspaper to wipe the surface area afterwards.....yup, an old black & white newspaper page as long as it's clean.......works better than paper towels. Just keep in mind that water or chemical "silicate" based deposits will not be removed unless a very aggressive pH program ( pH range of 1.7-2.5) was used which could wind up damaging the pool balls to become even worse looking and also totally unplayable. That type of program would require using acid and if I had the contents label of the product you used, I could probably recommend which type would work best......again this is a high risk type program when you use very low pH water.
Anyway, you don't have to worry about that with vinegar & water......I can't guarantee the results since I do not know what the product ingredients were that left the balls with surface deposits. But you have nothing to lose and this won't hurt the pool balls, however, the set will smell like salad for awhile. I suspect a couple of applications of Aramith restorer followed with Aramith cleaner should do a pretty nice job of restoring the sheen & eliminate any telltale odor of vinegar. I've previously worked in the chemical industry and I specialized in heat exchange equipment/systems which necessitated that water chemistry was a major part of my job focus.