Galton's interest in eugenics stemmed very much from his need to measure things - a foray first into statistics before he really looked into genetics and eugenics itself. He measured the efficiency of prayer, was the first to publish a weather map and use fingerprints in detective work. However, his observations of people afterwards related to his cousin Charles Darwin's theory of evolution then became the foundation of the study of eugenics.
However, there is no one gene responsible for any one trait that a person inherits. Height alone is regulated by at least fifty genes itself and they alone do not express all the variation observed in life. As the verdict now stands, both nature and nurture play a hand in the traits of a person and genes frequently adapt to the nurture, perhaps more often than the other way around.
For height specifically, there are several ways nurture can influence the gene. Should a person carry heavy things often or slump for prolonged periods of time, they will become shorter after time. For someone like me with scoliosis and a seemingly determined height of 5'5", exercise such yoga can eventually stretch out my spine so I become taller. The same can be said of a healthy spine as well.
Though what Galton observed was and still remains true, it is folly to think that genes can really be regulated. Not only do they change depending on the environment, they also mutate randomly by themselves, the faulty copying of the gene actually the driving force of evolution.