Can touching a skin wart and then touching the genitals cause hpv?

Warts are: caused by viruses, and yes they can spread by direct contact. But there are different kinds of viruses. You skin wart caused by a different type of virus than a genital wart virus.

Genital warts: Warts are caused by strains of the HPV, but skin warts are caused by a different strain than genital warts. Touching a wart on a regular skin surface does not spread it to your genital skin. HPV is a family of dozens of different strains, some producing various types of warts but most that do not. Each strain is very specific to where it is found.