Newton systems , with integrals of motion quadratic in velocities, are considered. We show that if such a system admits two quadratic integrals of motion of the so-called cofactor type, then it has in fact n quadratic integrals of motion and can be embedded into a (2n+ 1)-dimensional bi-Hamiltonian system, which under some non-degeneracy assumptions is completely integrable. The majority of these cofactor pair Newton systems are new, but they also include conservative systems with elliptic and parabolic separable potentials, as well as many integrable Newton systems previously derived from soliton equations. We explain the connection between cofactor pair systems and solutions of a certain system of second-order linear PDEs (the fundamental equations), and use this to recursively construct infinite families of cofactor pair systems.