The only way in which it could possibly be true is if you compared average standards of living. From the colonial period through to the late 19th century, the US population was composed of a majority of landowners. This stands in stark contrast to Europe, where land ownership was relatively quite rare, and had been for centuries. Most farmers in Britain were tenants (renters), and these people were increasingly moved off the land as England industrialized; it became more profitable for the landlords to consolidate farmland, especially for mass sheep farming. The dispossessed farmers ended up in the rapidly expanding towns and cities, where they in large part formed the new urban proletariat.

Now, it is true that most landowners, North and South, were small subsistence farmers who owned 100 acres or less. They produced very few cash crops, mostly concentrating on foodstuffs. This made them cash poor, but it bred a feeling of independence and self-reliance. More concretely, it also enabled a relatively high standard of living despite their "poverty," especially compared to the inhabitants of cities, who, in the 19th century, were far more prone to disease, bred by cramped and unsanitary living conditions, poor diet, and, sometimes, horrid work environments. Besides having greater access to a varied and plentiful diet, American farmers were much less exposed to disease in particular. This would come back to bite them when they were placed in urban-like conditions, where they generally suffered from disease even more so than urbanites due to their lack of resistance (2/3 of Civil War deaths came from disease).

Human height correlates pretty directly with social health. When conditions are good, when people are relatively free from disease and eating well, average height tends to go up, at least so far as genetics allow. When a population is stressed, especially by overpopulation, poverty, or other factors, it goes down.

If you compare estimates of average height for the United States with those of France or Britain in the 18th and 19th century, they are startlingly divergent. The British Army during the Napoleonic Wars boasted an average height of about 5'4" (among the enlisted; officers tended to be much larger than their men); the French Army, much larger and drawn from a broader cross-section of society, may have been marginally taller, but not by more than an inch or so. Both populations were as short as they had ever been, down from an early medieval average of perhaps 5'7" or 5'8". In contrast, Americans in the Union and Confederate Armies averaged slightly over 5'8", and I have seen speculation that the average in the late 18th century was around 5'9", scarcely less than it is today.