Could I catch leukocytosis by touching someone's skin?

No. Leukocytosis is not a disease. It is a medical term that refers to an elevation of white blood cells. White blood cells are produced in increased numbers when infection or inflammation is present.
No, usually not. Leukocytosis is an abnormally high number of white blood cells in the blood, as measured on a blood test. Both viral and bacterial infections can sometimes cause high numbers of white blood cells (wbc's) or low numbers of wbc's (leukopenia). Unusual leukocytosis, not caused by an infection, can be due to a bone marrow problem such as an early stage of leukemia. A doctor can further evaluate.