Is gravity a centripetal force? In curved space-time, there are curved paths. Since curved paths in our experience require some centripetal force to create them, isn't then gravity a centripetal force?
 A: The strong analogy is between gravity and fictitious forces. Of course centripetal force is the special case of rotating reference frames.
In Newtonian Physics you have $F=ma$, that is valid in an inertial frame. Suppose that we want to describe a particle that is moving with constant velocity, so $a=0$. If you want to study the same system in a non-inertial frame, than you need to add some fictitious forces: $$F=ma=0 \rightarrow F+F_{app}=ma=0$$
In general relativity, if the space is flat (no gravity) a particle moves on a straight line in space-time and follow a geodesic given by $\frac{d^2 \xi^{\mu}}{d \tau^2}=0$. If the space is curved, than the particle has, in general,  a curved geodesic, given by: 
$$\frac{d^2 \xi^{\mu}}{d \tau^2}=0 \rightarrow \frac{d^2 x^{\mu}}{d \tau^2}+\Gamma^{\mu}_{\nu \rho}\frac{d x^{\nu}}{d \tau}\frac{d x^{\rho}}{d \tau}=0$$
A: Insofar as the gravitational force points towards the 'centre of gravity' of an object (like the earth or the sun), it can be considered a 'centre seeking' or centripetal force.
