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.

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.

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

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.

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.

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.

I agree that zinc is one of the important micro-nutrients but people require a balanced blend of food for our biological engines.