It appears that folks like James C. Dobson of Focus on the Family, Tony Perkins of the Family Research Council, and Richard A. Viguerie are threatening a 3rd party bid if Guiliani gets the nod.
This is an empty threat. The religious right’s attachment to the Republican party is based almost entirely on hostility towards non-whites, and Rudy has the best bone fides in this area of any of the Republicans. The right wing Christo-Fascists’ followers will follow Guiliani because he has made an implicit promise to keep the black and brown man down.
The religious right’s entry into politics did not begin with Roe v. Wade, but rather it began with rulings that segregated religious schools could not be tax exempt 501(c)3 institutions.