We are currently building a home. Our builder operates as Inc, although it is apparent is a sole owner (one man band). Can we issue him a 1099?

Why would you want to issue him a 1099? Are this house being built as part of a business venture on your part? If this is a personal residence then no need for you as a customer to issue your contractor a 1099.

You are really confused, "he apparently operates as a sole owner"
Many, if not a large percent of corporations are one person, or husband and wife operations.
They are formed for tax purpose, not for size of company.
I for example have 3 corporations, two are just me, one for my rental properties, another for other investments.
Then I have one for my business which has a few employees.

The "INC" designation gives you the out for NOT issuing him a Form 1099-MISC.