Note that you can also define an infix operator by appending a subscript, e.g. /ᵣ.
This doesn’t seem crazy to me. It would ideally have to be done in Base to avoid type piracy.
You could alternatively define:
newtondivdiff(x::AbstractVector{<:Integer}, y::AbstractVector{<:Integer}) =
newtondivdiff(Rational.(x), Rational.(y))
and then just use / (which is exact for rationals or mixtures of rationals and integers).