I would like to argue in the opposite direction. The 17th century notation that is still in use today is a syntactic hodgepodge which does great disservice to students and their teachers alike, despite claims of its usefulness (by people who never tried an alternative). It is partially responsible for the fact that the average mathematician in the street cannot coherently describe the notion of a bound variable, thinks there isn't much difference between $f$ and $f(x)$, and is willing to believe that $\frac{\partial L}{\partial \dot q}$ is a sensible notation.

Functions as mathematical objects (as opposed to symbolic expressions) are fundamental to differential calculus. Moreover, important concepts such as derivative, definite integral, differential operator, gradient, etc., are themselves functions of *higher order* (they take functions as arguments).

Let me mention two modern foundational explorations in analysis.

First there is [Synthetic differential geometry][1] (introductory reading material [here][2] and [here][3]) whose distinguishing features are that it calculates with nilpotent infinitesimals, and that arbitary function spaces can be formed at will (whereas in classical analysis forming a function space is always a Big Thing). This makes certain definitions *very* easy. For example, the tangent bundle of $M$ is simply the space of function $\Delta \to M$ where $\Delta = \lbrace x \in R \mid x^2 = 0\rbrace$ is the space of infinitesimals (of order 2). And it does not even matter what $M$ is here, the definition just makes sense, both intuitively and technically. The classical approach to analysis requires a whole edifice just so that the tangent bundle can be defined. It is too complicated for the average undergraduate.

A foundation for calculus which is most directly based on functions is the [differential $\lambda$-calculus][4] (introduction [here][5]). The $\lambda$-calculus *is* the theory of functions. For example, functional programming languages are based on it. The differential $\lambda$-calculus is an enrichment of $\lambda$-calculus with (abstract) differential operators.

So, while I am sure somebody has cooked up a foundation of calculus based on avoiding functions, the arrow of progress points in the opposite direction.

  [1]: http://ncatlab.org/nlab/show/synthetic+differential+geometry
  [2]: http://www.math.uchicago.edu/~shulman/exposition/sdg/pizza-seminar.pdf
  [3]: http://publish.uwo.ca/~jbell/invitation%2520to%2520SIA.pdf
  [4]: http://www.pps.univ-paris-diderot.fr/~ehrhard/pub/difflamb.pdf
  [5]: http://iml.univ-mrs.fr/editions/preprint2001/files/difflamb.pdf