Yes, and many people live that way. I know of many people in recovery movements that have developed a personal sense of spirituality or connection to a higher power, that may be a force, or a being or some other concept that has nothing to do with a heritage book, or story, or description.
Belief systems go way back in human history. All primitive cultures have a way of dealing with the wonder one encounters when living in the natural world. For instance, Animism is really just respecting the life force of nature, and connecting how that life force came about and how we can each relate to it. These belief systems predate the Bible, the Koran, and the Bhagavad Gita.
And they avoid such minute detail that makes everyone else think you are a crackpot.