We could get to Mars with Newton, perhaps not as directly or efficiently. We could probably do it with vacuum tubes and slide rules, also. It's a matter of decimal places.
I imagine that you'd get to Mars in the end if you course corrected on the way and used more fuel than the most efficient trajectory. If you aim from earth using Newton you'll miss Mars orbit. I have this on authority from Radio Astronomer, whose job it actually is to calculate the orbits.