We show that the existence of the Newtonian limit cannot work as a selection rule for choosing the correct gravity theory from the set of all L=f(R) gravity theories. To this end we prove that stability of the ground state solution in arbitrary purely metric f(R) gravity implies the existence of the Newtonian limit of the theory. And the stability is assumed to be the fundamental criterion of viability of any gravity theory. The Newtonian limit is either strict in the mathematical sense if the stable ground state of a theory is flat spacetime, or approximate and valid on length scales much smaller than the cosmological scale if the ground state is de Sitter or anti-de Sitter space. Hence regarding the Newtonian limit a metric f(R) gravity does not differ from general relativity (with arbitrary \Lambda). That stability implies the existence of the Newtonian limit is exceptional to Lagrangians depending on R and/or the Ricci tensor but not on the Weyl tensor. An independent selection rule is necessary.

PACS numbers: 04.50.Kd