Can the result on cross section of some process be dependent on the renormalization scheme used?
If one calculates it to all orders, it shouldn't. In particular, if the coupling constant is small, the $O(g^k)$ accuracy calculation of the cross section obtained in different schemes can differ at most by $o(g^k)$, negligible amounts.
However, when one truncates the cross section at some order, the results may really depend on the renormalization scheme by subleading, higher-order terms.
Physically, cross sections are measurable so a valid theory must predict unequivocal values for it. And a renormalization scheme doesn't really "change the theory".