No! In about 21% of identical (monozygotic or MZ) twin pairs, one twin is right-handed and the other is either left-handed or ambidextrous. Since identical twins share identical genes, this is evidence that handedness is not a totally genetic trait.
Left-handedness is more likely to occur in twins than in single individuals. Only about 10% of people in the general population are left-handed. But about 17% of all twins are left-handed.
The cause of hand preference is not well understood. It is likely to be a combination of genetic and environmental factors. Environmental factors such as stress during birth, hormone levels during pregnancy, or position in the womb have been suggested. It has also been proposed that mirror-imaging in the formation of MZ twins could cause their differences in hand preference. Cultural influences also have an effect, as parents may encourage right-handedness.
Hand preference is a classic example of the controversy over the relative importance of nature (genetics) vs. nurture (environment) in causing a trait. A recent study examining the whole genome of nearly 4000 left and right handed twins was unable to find a strong genetic factor in determining handedness. Researchers concluded that it is likely that there are many weak genetic factors in handedness.