If melanoma is caused by a virus, does successful eradication of the disease through surgery happen because the virus is confined to the tissue in or immediately adjacent to the actual malignant lesions? Or is there usually a remaining level of virus in the body, but suppressed enough by the immune system to not become a problem in most cases?

As you might expect, there is a variety of scenarios here that can take place. The melanoma lesion tends to be isolated because it is a less common site of residence of the particular virus causing that cancer. That is why it does not bloom up everywhere on the body like measles or chickenpox. So the odds of a surgical removal of affected tissue that encompasses remaining virus are higher than many types of cancer, not to mention the surface location of the skin as the usual site of presentation makes a surgical extirpation less problematical than tumors arising in a deep site within the body. The problem is that sometimes the virus is located elsewhere in the body and will migrate to a new location, and if more vulnerable than average, perhaps for karmic reasons, the disease will reoccur even if a prior extirpation of tissue removed everything in the area that was contaminated with virus. This argues for use of a systemic antiviral regimen in all such cases, being less toxic and well-tolerated than body-wide chemotherapy or radiation.