Classical mechanics bears approximately the same relationship to calculus that calculus does to first-year algebra. You have to have a very good grasp of calculus to appreciate, or even survive, classical mechanics.
if you mean calculus as taught today --no-- newton did not invent that calculus -furthermore- in newtons book known as -principia- newton does not use -calculus- in any shape or form to -calculate- orbits - newton uses only proportions -- but feel free to downvote me for daring to -question- such a cherished -myth- instead of reading -principia- for yourself
http://ocw.mit.edu/courses/mathematics/18-01-single-variable...
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-c...
http://ocw.mit.edu/courses/mathematics/18-03-differential-eq...
It's a long road, but a very rewarding one. Good luck.