Simple answer: gravity *is* a centripetal force, and can be envisaged clearly as such in Newtonian mechanics.

*Centripetal* just means a force that is "radially inwards" ("directed towards the centre"). The electric force, for example, is also clearly centripetal. (It's slightly harder to define "centripetal" for the magnetic force.)

Your astronomy teacher is referring to [Einstein's theory of general relativity](http://en.wikipedia.org/wiki/General_relativity). His description is loosely an overview  of the topology (fabric) of space-time and how it interacts with matter/energy - the manifold is however 4-dimensional, not 3D.

In fact, test particles (particles which do not really disturb the gravitational field) in general relativity follow a [geodesic](http://en.wikipedia.org/wiki/Geodesic). This is effectively a generalisation of a straight line (shortest route) of normal Euclidian space to the curved space of GR, and may be seen as the source of centripetal force in Newtonian physics.