Yes Rita, they will. The VA hospital in Nashville, part of the Vanderbilt campus just excised mine and also a lymph gland and sent them to an outside lab for analysis. Thankfully there were clean margins and the lymph gland was negative. The derm department wants to see me every three months though for follow up.
They also removed a bladder cancer a year ago and they scope me yearly now to make sure it hasn’t come back.
I am however a Vietnam war era veteran. Not sure if a peacetime vet would have different benefits.

Hmm I’m a veteran- but peacetime. I wonder if they would cover my melanoma care?

It was zero problem for me as a Nam vet. Met with service officer at county courthouse. He did all the work. About six weeks later I had disability approved. KCVA has been incredible