Development of suburbia in the US is much more extensive than that and cannot be explained by "white fly" only. As a matter of fact, everyone lives in suburbia nowadays in the US. When visiting Atlanta, Martin Luther King's old neighbourhood consisting in apartment buildings was essentially a ghost town.
I believe the main reason is mostly economical. Historically, the US was an oil country with a strong oil industry promoting oil-powered transportation solutions. I believe that is the reason why cities in the US are so much car-oriented. Oil is abundant and cheap, individual transportation is more comfortable than public transportation, detached houses are nicer to live in than appartment buildings, so let's go for it and everyone should live this way.
On the other hand, Europe hardly produces any oil, realized it was a strong sovereignty issue in the 1970's and went for alternative solutions if possible. Therefore oil is more expensive, there is a stronger incentive to develop transportation alternatives less relying on oil (such as public transit), requiring sprawling to get better controlled.
Usually people believe that's a matter of space available, the US being so big as a country and having so much room. But when you look at it in details, the US population is very unevenly distributed, and in the Eastern half of the country, the average population density is the same as in Spain. Yet Spanish cities are considerably denser. Obviously older cities built essentially before development of cars are denser than cities which developed afterwards, but then again, even very old US cities like Boston or Philadelphia sprawled much more than let's say Berlin, which actually developed later.
In the past 50 years, the most ambitious urban transportation projects in the US were mostly car-related (such as Big Dig in Boston). On the other hand, Spanish cities developed very extensive metro networks during the same period. The core European problem is that the more you need oil, the more you're dependent of countries like Saudi Arabia or Russia.