It is widely held that ZFC cannot be shown to be self-consistent or complete.
So, what happens to the system of calculus? Can it be shown to be complete or self-consistent? (Edit: Oops. So, two questions here: Can the system itself prove its own consistency and completeness? And what system of logic can prove its consistency?)
By calculus, I mean univariable/multivariable integration, differentiation and limit.
The commonly used system for real analysis is certainly strong enough to express Peano Arithmetic and thus suffers from all the Goedel incompleteness issues. This thus pretty much answers your question but I think some more information might place things in a wider perspective so I added gratis.
The portion of calculus you mention is mainly relying on the model $\mathbb R$ or real numbers. Very rigorous models of the real numbers can be constructed from models of the rationals, which in turn can be constructed from models of the integers, which in turn can be constructed from models of the naturals, which in turn can be constructed from any model of $ZF$. The latter can't be proved consistent (unless it is inconsistent).
There are other possibilities for modeling analysis for instance using topos theory. There are toposes that include a real numbers object and analysis can be done there. One of those toposes is the ordinary topos of sets and the real numbers object there gives rise to the standard notions of real analysis. There are other toposes, for instance there is a topos with a real numbers object in which every function is continuous (thus realizing Brouwer's intuitionistic dream). Other toposes exist as well with various (sometimes weird) theories of analysis.