The answer really depends on definitions, as is so often the case.
Here is a definition from a University of Minnesota website that I find to be reasonable:
Spirituality is a broad concept with room for many perspectives. In general, it includes a sense of connection to something bigger than ourselves, and it typically involves a search for meaning in life. As such, it is a universal human experience—something that touches us all. People may describe a spiritual experience as sacred or transcendent or simply a deep sense of aliveness and interconnectedness.
By this definition, the Overview Effect is definitely a spiritual experience, in that most astronauts do have an understanding that they are part of something greater than themselves. As they gaze at the Earth, they usually stop identifying with places on the Earth, like their hometown, and begin identifying with the planet as a whole.
As they look beyond the Earth, some astronauts begin to have an affinity for the universe as a whole. This shift from parts to whole is often reported in Earthbound spiritual experiences. The main difference is that while in orbit or on the moon, weightless or in a reduced-gravity environment, the realization may occur more rapidly.
So, in this sense, the Overview Effect is spiritual in nature, but a great deal depends on the individual and their prior experiences. Apollo astronaut Edgar Mitchell told me that everyone had the same experience, but everyone also interpreted it differently.
Of course, this is true of Earthly spiritual experiences as well, but we always come back to this sense of the whole and of being part of something larger than ourselves.
(c) Copyright, Frank White, 2016, All Rights Reserved
The Overview Effect: Space Exploration and Human Evolution is available at aiaa.org and amazon.com