In the narrow statistical sense, a conditional probability refers to a probability that depends on the outcome of another random variate. The posterior probability, however, depends on the entire prior probability distribution p(θ), not on a particular outcome θ. So, I would say strictly speaking: no.
This is, unless you see the prior itself as a random variate, which would be possible to argue I guess, albeit somewhat unusual. This reminded me about a an older discussion between Larry Wassermann, Andrew Gelman and others, on whether the p-value is a conditional probability, and if it is appropriate to write p(d>D|H0).