Disclaimer...I don't sell this any longer.
One of the many products I used to sell as a distributor rep was Eximo for turf. Some of you guys might still use it today. The base ingredient was SynTech, a synthetic acid. The manufacturer is Environmental Manufacturing Soulutions (EMS). EMS had another product I used to sell called Barracuda. Barracuda was designed and sold to dissolve concrete off ready-mix concrete trucks by dissolving the calcium in the lime which is a major ingredient in concrete. It also happened to do a great job cleaning carbon build-up from exhaust on greens and fairway mowers, and the grass stained build-up you get on your cutting units, too. I was showing it to a mechanic one day and he asked if we had anything to remove the iron stains from his spray tank, sprinkler heads, etc as they were on a high iron spray program. I didn't know if EMS had such a product so we tried the Barracuda. It worked wonders on the spray tank. Spray it on, let it sit until almost dry, and wipe clean. Spray tank looked like new again. It also did a great job removing iron stains from granite tee monuments.
Barracuda would probably do the job for you, but if the patio white stone is a limestone base material it will dissolve some of the calcium in the stone. It's not like it will turn your stone to mush, but if there is calcium in your stone it will dissolve a small amount of calcium with each application, but it will take the iron with it. If it's more of a granite type material like on some tee monuments it will safely take the iron stains off without damabing the granite.
http://www.enviromfg.com/products_barracuda.htmGive EMS a call and see what product they might suggest.