Calculus is not a description of the physical world, but a description of an idealization of the physical world. In that idealization we regard space and time as a continuous space. Whether space and time are actually continuous -- or, for that matter, whether the universe is finite or infinite, bounded or unbounded -- is essentially irrelevant to calculus.
On the other hand, one might ask a slightly different question: If the universe is discrete at the Planck length, would Calculus still be useful for describing it, and if not, would there be a more suitable alternative? To this question, the answer is: Certainly Calculus would still be useful for describing the universe at scales much larger than the Planck length, because at those scales the discrete nature of space and time is essentially imperceptible. As one approaches those very small scales, the error in modeling space and time as a continuum would begin to mount, and predictions made by the idealization would stop matching up with reality.
But there is no need to ask "What if?", because in fact physicists have been dealing with this for decades. Much of quantum field theory is done using a model of the universe in which spacetime is discrete at very small scales; this is known as an ultraviolet cutoff. The name comes from the fact that small lengths correspond to high energies and frequencies; similarly, a model of the universe in which spacetime has some finite size (maximal lengths) is called an infrared cutoff, because large lengths correspond to low energies and frequencies. There is a short article about both cutoffs on Wikipedia and there are a number of questions about them on the Physics Stack Exchange site.