It would be EXTREMELY unusual for ice to give you a burn of the type you describe. 10-20 minutes of ice directly on the skin without protection wouldn't even do it. It is more likely related to your surgery. Possible an effect from the surgery itself or possibly a superficial skin infection (Staph/Strep). Avoid any UV exposure over time to minimize pigmentation. But, first and foremost make sure you see your doc to discuss and make sure not serious.
Not likely from ice.