Why don't black and white Americans live together?

Partially because of Fannie Mae and Freddie Mac and the Community Reinvestment Act, banks were incentivised. Black families were targeted by predatory lenders who knew the rules well enough to get the loans for the individuals, individuals who perhaps were not in an economic position to be able to fulfill the agreements of the loans, then resell them in mortgage backed securities. At the micro level, the Community Reinvestment Act was "a good thing". We as a society want everyone to have a home. However, it did have negative consequences at the macro level.

So there was a business opportunity at one time. Is there today? I don't know. And yes, it was probably an over-characterization.

White migration does happen, and we call it gentrification, because it usually means wealthy young (mostly white) folks moving into poor neighborhoods on the edges of "cool" cities, causing rents to rise dramatically over a short period of time, and pushing out longtime residents who are too poor to afford to continue to live in the hip new neighborhood with fancy bistros and cat cafes (yes, there's really a cat cafe...it's in East Austin on the site of a business once owned by a Hispanic family whose business was bulldozed).

Integration almost has to go the other way, or it is more of a disruptive force, and a force for further damaging communities of color. Race and class are so closely correlated in the US that you can't talk about one without the other on most issues. In this case, it means that white people moving into a black or brown neighborhood are often displacing black or brown people who can no longer afford to live there, because wealthier whites have driven up rents (and home ownership among black and brown people is much lower than ownership among white people in the US, so that doesn't provide a very good buffer to protect those black and brown neighborhoods from being "flipped" into white neighborhoods).

In short: Integration is a good goal. But, without some care, moving white folks into black and brown neighborhoods can be harmful.

Interestingly, Asian-American integration seems to happen only within higher socioeconomic classes. Asian-Americans as a group seem bifurcated into South or East Asians who immigrated into highly-technical, well-paying jobs after 1965 (when U.S. immigration policy opened up to let them in), and East Asians who came over for things like the California railroads in the late 1800s or Southeast Asians who arrived as refugees in the 70s and 80s. The latter group is much more likely to live in Chinatown, work in restaurants or laundromats rather than software companies, have few friends of other races (indeed, inner-city Asian-Americans I knew in college seemed more likely to hang out with the black kids than the white ones), and speak relatively poor English.

This makes me think the root cause is economic. It could have to do with these groups not being able to afford houses in nicer areas, or it could be people not wanting to associate with people who they view as a rung down on the socioeconomic ladder.

You're implying that they don't have the choice of an equally posh black neighborhood, in which case such a slight preference against being a minority wouldn't be surprising at all.