For a recognizable U.S. as we know it, there are two ways in my opinion.
First is the Patriots conquer Canada in 1775. The addition of Quebecois Catholics from the start could do much to prevent the development of the stigma, particularly if Washington selects Charles Carroll as his VP later on, as was apparently considered in 1792 IOTL. This would get the U.S. at large used to such from the start, when things were still being defined and social norms stated in the Post-Revolution period. Later on, the U.S. could continue it's expansion and add Catholic Mexico and Central America to itself. Such, combined with Catholic immigration from Europe, could result in a majority Catholic America.
The second, and admittedly this is more a rip off Decades of Darkness, is something I've toyed with lately. The Russians achieve a decisive success at Berezina in late 1812 and the Napoleonic Wars are brought to a close by the Summer of 1813. Free of the European distraction, Britain is able to fight the war in North America to a clear victory with additional resources. New England is set up as an independent Federation, while the British take Michigan and Wisconsin, setting the border from thence on the 46th Parallel North to the Columbia River. This strips off a lot of Protestants and OTL Protestant immigration; Illinois, Indiana and Missouri will become firm Slave States here. Over the rest of the 19th Century, a more Southern-dominated U.S. is aggressive and expansionist, annexing Mexico over the course of the 1830s and 1840s. With fewer Protestants and a lot more Catholics, same outcome as earlier; probably sufficient to get a majority here.