I don't know how it's taught in other places, but here in Brazil I was taught in school that Newton developed Calculus - an entire realm of Mathematics - because there were no other tools he could use to describe the natural laws of physics. However, considering what we know about ancient civilizations' use of mathematics, engineering and astronomy, I find it very hard to believe that they didn't have their own versions of advanced maths.
My question is: do we know of Calculus popping up in History before Newton?
While we wait for an answer that expands on what it was that Newton (and Leibniz, not to forget him) actually invented, you might find this older answer (by /u/ManicMarine ) and the follow-up discussion interesting.
In brief, many problems which we today would consider to be calculus could be solved numerically by ancient mathematicians from many different civilizations. What Newton and Leibniz actually did was to introduce a new formalized way to work with these problems.