You've got to be careful here to distinguish between Newtonian gravitational mass (gravitational "charge") and Newtonian inertial mass (measure of inertia).
Could there be, in Newtonian mechanics, a particle with non-zero inertial mass but no gravitational mass? I think the answer is, in principle, yes. It would simply be a particle that does not gravitate.
Could there be, in Newtonian mechanics, a particle with no inertial mass? Such a particle would have zero momentum always (unless you allow for actually infinite speeds...). Since, the particle's momentum is constant, by $\vec F = \frac{d\vec p}{dt}$, there is no, and can be no, force acting on the particle; the particle is a "ghost".
So if, by photon, you mean a gravitationally massless particle then a Newtonian photon doesn't gravitate.
If, by photon, you mean an inertially massless particle, then a Newtonian photon is a "ghost".