Take our universe. Observations are consistent with relativity, but not consistent with Newtonian mechanics. Assume that our current (relativistic) model of gravitation is correct. Now increase $c$ to infinity. Experimental observations in this altered universe are still consistent with relativity (by the assumption that relativity is correct). Are they now consistent with Newtonian mechanics as well?
In terms of special relativity versus Newtonian mechanics, the answer is yes. However in terms of general relativity this would be a much different universe. Instead of having curvature in a 4 dimensional Minkowki space-time, We would have a 1 dimensional time and a 3 dimensional space universe with no possible connections between space and time. So I don't think the equations of general relativity would result in the kind of gravitational forces we experience in our universe.
A better definition of an 'infinite speed of light' is needed. If by that you mean that observers in relative motion will view the same space up to isometry and observe equal passages of time between events, then yes. However, there is no such thing as a 'relatavistic model' in Galilean space-time.
For example, electrodynamics is obviously Lorenz invariant. If you take as axioms Coulomb's law of electrostatic force and the Lorenz invariance of electric charge, you can derive the magnetic field (the Lorenz force may be derived from the Coulomb expression of force by Lorenz transformations since force is not a 4-vector).
However, if you replace Lorenz invariance with Galilean invariance in special relativity, all you have is Coulomb's law for electrostatics equipped with the fact that it also works for moving charges. This sort of physics, however, is really just like Newtonian gravity but without the principle of equivalence.