So, this question is a little difficult to answer, at least as phrased. The nineteenth century was a long time, and the United States underwent drastic changes throughout, going from 16 states that were separated by forests, mountains, and a relative lack of roads, canals and other internal improvements to 45 much more closely integrated states joined not only by an extensive network of railroads but also the instantaneous communication of the telegraph. The country had engaged in wars of conquest against Mexico, Spain, and a rather large number of American Indian nations. The Census Bureau had announced the so-called closing of the frontier in 1890. In general, the United States had gone from a collection of agrarian former colonies whose independence from England was still an open question to at least a second-tier world power with an empire of its own, backed by one of the largest economies in the world. By 1900 the genocide and forced relocations of American Indian peoples was more-or-less complete, meaning that opportunities and incentives for any sort of intermarriage were greatly reduced, or at least changed.

That said, yes, in the early nineteenth century there are many accounts of intermarriage between whites and Indians, both in a more "Indian" context, where white people (mostly men) lived among indigenous groups, and in a "white" context, where indigenous people joined white settlements through marriage. One of the most famous early works of American literature, James Fenimore Cooper's The Leatherstocking Tales takes as its main character Natty Bumppo, a white man who spends considerable time among and has adopted the mores and folkways of American Indians (as they were understood by white people).

That said, the nineteenth century saw much less of this than previous centuries, as by 1800 both the United States and Canada were well on their way to developing settler societies based on agrarian trade and even industry. Canada less so, perhaps, but I bring it up because many of the white men who intermingled with Indians in the 17th and 18th century were actually of (French) Canadian extraction rather than (Anglo) American. They were voyageurs, engaged in the fur trade, either trapping beavers and taking their pelts to trading posts to exchange for food and trade goods, or trading with Indians involved in the same activities.

Nineteenth century Americans, like all good white people of the time, were obsessed with racial identity and classification, and the idea of these people who intermarried with Indians was a powerful one. There was a lot of worry about miscegenation not just with African Americans, but with Indians as well, and so the captivity narrative became a common trope. In this type of story, sometimes based in truth but usually heavily embellished, a virginal white woman is taken captive by savage Indian men and often taken advantage of sexually and/or killed, until hopefully rescued by a virtuous white man. Art of the nineteenth century is obsessed with these stories, as in the infamous capture and murder of Jane McCrea during the Revolution. This incident was adapted by Cooper for The Last of the Mohicans, and the subject of a famous painting by John Vanderlyn.

So, to sum up: there was some cultural and sexual mixing between white and Indian people in the nineteenth century, but much of this was a legacy of an older era and would soon become relatively unheard of, especially during the period of mass emigration across the Plains after the Civil War, when the United States Army was waging a genocidal war against large numbers of Native Americans. Even after the first Indian Removal (often known as the Trail of Tears), intermarriage became much less likely.