Can I terminate an employee for visible tattoos?

Generally, yes, tattoos do not fall within a "protected category" (from discrimination).

Best practice, to avoid legal challenges, would be to institute a policy relating to visible tattoos in the workplace - like any other dress or grooming policy. Furthermore, apply the policy consistently without regard to race, age, sex, or any other protected class.

One issue to consider is when the tattoos are for a religious purpose. Although this is not a frequent occurrence, should the employee claim the tattoos are religious, the employer should attempt a reasonable accommodation. This could mean a transfer to a less visible position, but again, be cautious not to discriminate.