As a decidedly amateur historian with interest in this area, I think that claim is true on its own but also does not tell the entire story.

It is true that decentralization was viewed as a major component of nuclear survivability, particularly in the earlier part of the cold war. This was the time period during which the FEMA (today's name) crisis relocation program was being devised, for example, and the high cost and complexity of crisis relocation (which, in an earlier form, was a major motivation for the Eisenhower freeway system, to the extent that it's probably fair to say that it was the main motivation with materiel movement as a second) served to highlight the inherent vulnerability of dense cities and keep it very much in the minds of government planners. Decentralized cities had an inherent advantage to planners in that money could be saved on crisis relocation efforts. Of course the crisis relocation program was never fully implemented, but the way of thinking was fairly influential.

The federal government had an enormous role in suburbanization of US cities in many, many ways, which is actually part of what makes it hard to address this point. Support for suburbanization was not coming just (or even primarily) from FEMA, all kinds of federal agencies had a hand. Much of the urban renewal work of the 20th century took the effective form of relocating poor people to the suburbs and replacing their inner-city housing with industrial/commercial/transport, for example the Model Cities program of the late '60s. This was in part a result of the general feeling that the inner city was where poor people lived and so improving their economic situation required getting them out of it, part of it was merely the practical issue that substantially improving a dense area is a lot more expensive than razing it and build something new there. I don't know if these programs were strongly influenced by crisis planning, they probably were at least in part, but it seems unlikely that crisis planning was a much bigger influence in federal advocacy of suburbanization than the more organic trends of white flight and urban decay that came out of a set of race and class relations, in a potent combination with some simple budget and timeline considerations.

My point is that nuclear planning was a factor, but the massive suburbanization of the postwar decades originated from many factors out of which nuclear crisis planning was only one, and I'm not convinced that it was one of the bigger ones in the end. Yes, the Bulletin of the Atomic Scientists very openly advocated for suburbanization of cities and that view was influential, but at the same time so much of suburbanization was motivated by a new vision of the American dream that came out of the peculiarities of post-war economics and demographics, racism, the radical popularity of the automobile (which not only enabled low-density areas but often required the destruction of high-density areas to provide freeway access to business districts), and probably at least a few other things.

Any claim that "urban sprawl is a result of x" where X isn't a list of things is probably pretty hard to defend. A complete change in not just urban planning but people's patterns of life tends to require a confluence of factors.