There are lots of Indian influences on modern American culture, but no, it doesn't really compare to Mexico. Honestly, it comes down to a combination of numbers and politics.
If you have a look at the Historia verdadera de la conquista de la Nueva España, Bernal Díaz del Castillo's account of the Cortez expedition, it's clear that (1) the Spanish conquest only happened in the context of a general revolt against Aztec rule and (2) the Spanish were vastly, hugely outnumbered by Indians the entire way. In addition, the Spanish took political control early and kept it. Even after disease annihilated the Indian population-- most estimates put the death rate at over 90%-- the political conditions existed for intermarriage between Spanish and Indians, a process called mestizaje. (Mann's 1491 has a very good discussion on death rates.) Emperor Moctezuma II's descendants provide a useful example of this. Moctezuma's daughters married Spanish nobles, and Moctezuma's great-grandson, Pedro Moctezuma de la Cueva, was made a count by the King of Spain.
That's rather different from what happened in the U.S. I can't speak for Canada, which is outside of my knowledge. The Indians in this country were equally devastated by diseases, as far as we can tell, but there were two major differences.
First, there were more settlers than Indians in what became the United States. Way more. By 1660, the colonists in New England outnumbered the natives 6:1, through a combination of immigration, high birthrates, and low infant mortality. (See Jack Greene, Recent Developments in the Historiography of Colonial New England, for more details.) By the 1830s, this ratio had only gotten worse for the Indians. There were roughly 70,000 Indians in Mississippi, Alabama, Georgia and Tennessee who were removed to Oklahoma, against 1.6 million Americans in those four states, out of a total U.S. population of 12.8 million. I don't have the breakdown by race in front of me, but you get the point. EDIT: There were ~540,000 slaves in those four states, which gives us a total of 1.06 million white settlers. That's 15 to 1!
Second was the legal policy of the Powers that Be. The general feeling of whites in the United States was that Indians would have to be civilized as individuals rather than as tribes. As such, before 1924, the Indians were treated as members of independent nations who were not entitled to American citizenship unless they were to leave their Indian-ness behind and be naturalized as American citizens. Elk v. Wilkins, 112 U.S. 94 (1884). This is very, very different from the Spanish policy, which had recognized that the Indians were Spanish subjects (at least in theory!) and subject to the protection of the Crown. Las Casas' Brevísima relación de la destrucción de las Indias, and El amanecer de los derechos del hombre, by Dumont, are good reading on this.
The premise of your question is not valid for all of North America.
There is a thriving Métis culture in Canada, a wonderfully complex blend of Native and Hispanic traditions in New Mexico and throughout the greater Southwest, and the persistence of Native American place names throughout the continent. Lacrosse is the official sport of Canada, and played by over 750,000 people in the United States. We eat Native American plants like strawberries, cranberries, corn, tomatoes, peppers, squash, all manner of beans. We smoke tobacco, we paddle canoes and kayaks, and many of our interstates follow original Native American trade pathways throughout the continent. There is a deeper Native American legacy written into the fabric of North America than most realize.
That said, the heart of question has more to do with how the concept of race in the English colonies, inherited and transformed by the new United States, influenced the perception and acceptance of Native American traditions compared to the lands claimed by Spain and France.
While españoles occupied the top rung of the Spanish casta system, the culture of the empire made a place for multiracial subjects. Different social mobility, taxation, privileges, and economic opportunities were afforded to individuals based on ancestry, but even within the system socioeconomic status could be negotiated, particularly along the frontier. This admixture and stratification of genetic diversity is still written in the landscape of genetic diversity in modern Mexico. The French were always greatly outnumbered by their Native allies, and the growing Métis culture represented a means of marrying into Native American kinship systems to harness existing trade networks and expand influence for trappers and coureur des bois.
As Calloway states in One Vast Winter Count, the British Empire was hierarchical and authoritarian, but ethnically inclusive. Native Americans were accepted subjects of the crown, desperately needed allies in the game of empires against France and Spain, and could regularly lobby for their interests through official diplomatic channels. The new United States, however, was "egalitarian, democratic, and ethnically exclusive", reserving citizenship to free white persons. Where other colonies made room for admixture, the emerging concept of race in the United States forever excluded African Americans. Native Americans "were eligible for inclusion if they gave up the things that made them Indian and adopted white American ways of living and thinking" (p.428). While other empires in the New World embraced, or at the very least tolerated, integration of Native American traditions the culture of the United States devalued and stigmatized Indian influence. The Discovery Doctrine stripped Native Americans of sovereignty over territory, and a land-hungry nation, empowered by the entitlement of Manifest Destiny, justified expansion into inhabited lands. In many portions of this country we inherit a landscape relatively free from the trappings of indigenous life, all the while not realizing the exclusion of that influence was actively cultivated by a country justifying it's expansion.
That said, admixture occurred, and the patterns of modern genetic ancestry of the United States tell the messy story of populations shifting and changing in the five centuries since contact. There is much more to learn about the history and story of North America. I hope this helps to answer, at least in part, a very complex question.
