This is a huge, and interesting, question. I'm afraid our current data can only go so far in providing an answer. What follows is an analysis of the data, and the pitfalls of trying to definitively answer this question due to everything from notions of identity, evolving racial categories, and lack of solid sources. I will first address the research surrounding Native North American population size at contact, then the issues with calculating the rate of decline until the numbers hit a nadir around 1900. I will then address historical census numbers for the European and African descent populations of the United States. Finally, I will emphasize how all of this is simply an educated guess, based on flawed sources, that reflects a dark history of our nation’s interaction with Native Americans.

First, gallons of ink have been spilled debating Native American population size at the time of contact. There are two main camps, “high counters” and “low counters”. In the early 1900s scholars like Kroeber and Mooney looked at the Native American population size during their time, and assumed little changed in Native American lifeways between contact and 1900. They didn’t factor in mortality from disease, warfare, and famine. Kroeber estimated 900,000 people lived in North America, and 8.4 million lived in the entire New World. That is a population density of less than 1 person for every six miles. Other scholars, like Dobyns, formed a group called “high counters”. The high counters held the New World was richly populated at contact, but catastrophic disease spread ahead of colonists, decimating the population, and rendering any colonial-period estimates of population size grossly inaccurate. Dobyns estimated over 112 million people lived in the New World at the time of contact. For reference, only 11 countries today boast a population larger than 112 million.

Today the popular perception has inherited the legacy of the high counters, and their catastrophic, apocalyptic population decline due to infectious disease after contact. In academic circles we are stepping back from the assumption of epidemic disease decimation without concrete evidence of that disease mortality. For example, ten to twenty years ago we might look at a Mississippian site abandoned around 1525 and assume disease carried off all, or at least most, of the inhabitants. However, we now know for many people in North America geographic mobility was a regular means of dealing with resource scarcity, or territorial encroachment, or changes in the political structure. The interpretations of the evidence have changed, and with that change we must modify how we reconstruct the past. All that said, current best evidence suggest the Native North American population size at contact was between 2 and 7 million people.

As you can see, there are a host of issues and assumptions that go into calculating the Native North American population in 1492, and those issues are only magnified in the years following contact. Native American population size, and subsequent decline or rebound, varied based on geographic location and time. Centuries of warfare, population displacement, epidemic disease mortality, famine, territory restriction, and identity erasure wreaked havoc on Native North Americas. Calculating raw numbers is next to impossible. Native Americans were not included in the U.S. Census until 1860. An attempt at a total count of Native Americans throughout the U.S. was not realized until 1890. Compiling first hand estimates given by those traveling into Native American lands can only tell us so much from their limited perspective. Scholars generally agree the Native American populations hit their nadir (lowest point) by 1900 when the U.S. Census total for American Indian, Eskimo and Aleut was 237,196 (roughly 0.3% of the U.S. population). Since the population decline debate is so contentious, I’m just going to focus on when the “White” population of the United States passed the upper bound of the best estimates of Native American population size at contact.

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I think Thornton's [American Indian Holocaust and Survival: A Population History Since 1492] (http://www.amazon.com/American-Indian-Holocaust-Survival-Civilization/dp/080612220X) would probably be a good resource for answering your question (it's at my work office atm, so I'm reluctant to give a more definitive answer to your question lest I answer it incorrectly.)

Clearly, the shift didn't take place all at once - it would have been somewhat gradual, from geographic region to geographic region, as peoples came into contact either with one another and got into it over land, or (as in the case of the Columbia River region) came into contact with diseases that decimated their populations before they ever even saw a non-Native. Lewis and Clark's journals report coming into contact with the [Clatsop] (https://en.wikipedia.org/wiki/Clatsop) at the mouth of the Columbia in 1804 as an [already decimated people] (https://franceshunter.wordpress.com/2009/09/21/lewis-clark-and-smallpox/).

The Dakota definitely seemed to be aware of their being a population tipping point for them getting outnumbered, and that that was a very dangerous moment: [Thaóyate Dúta] (https://en.wikipedia.org/wiki/Little_Crow) repeatedly warned his people, other Dakota leaders, and non-Native leaders in Minnesota Territory, that the displacement of the Native population by White settlers and the subsequent loss of Native lands would lead to war. That's [exactly what happened] (http://www.usdakotawar.org/), with dire consequences for the Dakota that have repercussions to this very day.

There are certainly [parts of the United States where it's still not true that Natives are outnumbered] (https://en.wikipedia.org/wiki/List_of_U.S._communities_with_Native_American_majority_populations), but I understand that you mean the US as a whole.

Edit: [Here's] (https://s-media-cache-ak0.pinimg.com/736x/f7/76/be/f776be148f2f41ab4ba6988b0b22550b.jpg) a cartoon about this exact situation that's so relevant I had just assumed it was by Dine cartoonist Jack Ahasteen, but - nope, it's by non-Native cartoonist Matt Bors. I remember when this cartoon came out, it really struck a chord with all of us it was so on-point.