since America is one of the only places that uses CDMA for cell phones, it makes it a uniquely American problem.
Wrong. Qualcomm's patents definitely extend past their proprietary IS-95 and CDMA2000-based systems. 3G GSM using WCDMA and 4G LTE all depend to some extent on Qualcomm's intellectual propertly.
Sure, to use the OP's example, Samsung pays a lot less to Qualcomm by developing their own 3G/4G modem chipsets rather than sourcing them directly, but it's not like Qualcomm was stupid enough to cut themselves out of the game completely. In short, they aren't going anywhere for a good long while even after CDMA2000 is a page in the history books.