Can a full time Religious Worker on R-1 in the US engage in private
business in addition to his religiuos activities without violating his
status?

Greetings,
This is, of course, no legal or professional advice, and also only a
free comment, NOT an official answer to your question.

This US government site gives the definition of a Religious Worker,
including this passage:
"The applicant is entering the United States solely to carry on the
vocation of a minister of that denomination, or, at the request of the
organization, the applicant is entering the United States to work in a
religious vocation or occupation for the denomination or for an
organization affiliated with the denomination, whether in a
professional capacity or not; "

http://travel.state.gov/visa/temp/types/types_1272.html

This would seem to indicated that private business activities would
not be allowed:  "...is entering the US solely to carry on ...".
This would also seem logical, since persons with an H1B visa for
temporary employment are not allowed to work for another organization
than that which has applied for their visa.

As said, this is not a final answer to your question.