Countries with strong native cultures like Mexico, Peru, Paraguay and Bolivia have large mestizo and native populations. Bolivia in particular has the largest Amerindian population if I'm not mistaken. By the time the 13 Colonies started to agitate for independence, native numbers in North America were way too small to have a major mestizo population, let alone native majorities. When/if mass inmigration starts, it would become an even lower proportion
You could have a massive push by the Founding Fathers to adopt native culture in government (like they considered themselves heirs of the Greco-roman tradition in many cases) but it's unlikely to be very accepted by the populace; in fact many colonists were hostile to the natives and wanted the revolutionary government to allow them to settle native land which it eventually did. Even if sucessful it would be native culture as seen by the lenses of European colonists. You could have a more vibrant mestizo culture in the US (though I'm not certain how), but they would never amount as a strong segment of the population like in Latin American countries. It might be possible to have the Civilized Tribes to reach an elite status and be dissproportionately influential in US culture and politics, but I'm not sure how and it's not very probable. After the 1700s I'm not sure it is possible, with the constant wars against Native Americans and Manifest Destiny adopted widely through society.