No, a Newtonian universe with finite speed of light is consistent. People were able to do physics in that model for two centuries. Special relativity flows from

No, a Newtonian universe with finite speed of light is consistent. People were able to do physics in that model for two centuries. Special relativity flows from the much stronger and more radical assumption that the speed of light is constant in all frames of reference, which is what the Michelson-Morley experiment showed to be true in our universe.