A rich diet does lead to earlier puberty but to my knowledge it does not student the adolescent growth spurt. Early pregnancy, before reaching adult height, does slow or stop growth.

Young children from Guatemala who are adopted into the United States gain considerable height relative to the standards. So the height difference cannot be genetic.

The English were relatively short in the mid-nineteenth century and so their expeditions return with stories some exaggerated, not doubt about "giants" who lived in other parts of the world.

Genes are very important at the individual level but these differences tend to cancel at the national level.

Being shorter or taller depends on details of the diet. Vegetarians can obtain a balanced diet, with sufficient complete protein, vitamins, minerals and micro-nutrients. It's trickier for vegans but it can be done.

Africans are not as tall as northern Europeans. But Manute Vol and other sports figures in basketball and football receive a lot of press coverage.

The second and third generation effect of immigration accounts for only a small portion of the height shortfall between Americans and northern Europeans. Besides the slowdown in American heights began in the 1950s, well before large-scale immigration into the country.