The short answer is yes, you can and should teach the Bible in public schools.
The long answer is more nuanced.
There are three subjects that benefit from the inclusion of the Bible: English, Social Studies, Political Science, Western Law, Art, Music and yes, even Science.
We expect students to recognize and understand literary allusions. The vast majority of literary allusions come from four sources: the Bible, Shakespeare (who often alludes to the Bible), Greek mythology and popular culture. There is no good reason to deny students understanding of certain literary allusions, merely because they come from the Bible. The Bible is also a literary classic in its own right. Belief is not a prerequisite to an intellectually honest presentation of the Bible as literature.
Avoiding the Bible also leads to miseducation, such as the case of a fifth grade teacher who defended reading The Lion, the Witch and the Wardrobe by C. S. Lewis, by saying she intended to read it as a fairy tale. C. S. Lewis intended the story to be Biblical allegory, not a fairy tale. To teach otherwise is educational malpractice. Either the teacher should teach literature such as this honestly, or avoid the book entirely. The middle ground simply will not do.
History education prefers primary sources whenever available. The Old Testament is the major primary source for the ancient history of the Jewish people. The history of the church had a huge impact on the history of Europe over the last 2000 years, and an understanding of the Bible informs our understanding of European history. The Boers drew their rationale from the Bible (although I would argue that deliberately or not, the Boers improperly applied the Bible to their situation). In fact, an understanding of the Bible is essential to an understanding of the motivations behind many historical events.
Political Science and Western Law:
Our public discourse constantly refers to the Bible, and yet most of the people who think they are quoting the Bible (both Christians and non-Christians alike) have near zero understanding of the Bible context itself or the Bronze Age time when most of it was written. Christians especially have a weak understanding of what a “literal” interpretation means. When I was much younger, I met a man who had been an Air Force pilot during WWII. After the war, he went to Papua New Guinea or Irian Java (I forget which) to be a missionary. The island people had a noun which meant airplane. Literally, the word meant “a bird with the skin of a machete.” We would be foolish to think that the island people really thought the airplane was a bird, yet Biblical “literalists” make this type of mistake all the time. Another example comes from Chinese. Their word for computer means “electric brain,” but clearly the word is not figurative, spiritual or symbolic. It is simply the word for computer. In English, we still say the sun rises and sets, but no one supposes that we literally mean the sun moves up and down. Many people who say they believe in taking the Bible literally fail to distinguish these types of expressions, leading to some of the ridiculous arguments we hear everyday.
As Christopher Gunter wrote :
So what are young people to think when they hear biblical passages taken out of context to both support and refute gay rights, or the Iraq war, or any highly charged issue? They must not be afraid to question and challenge biblically based sound bites. They must have the courage and the foundational knowledge to understand for themselves the source and context of biblical passages. Our reluctance to teach the Bible perpetuates its mysteriousness, which has grave consequences in our intellectual lives and in the wider world in which we live.
Art and Music:
Anyone who study art or music appreciation will not get very far before they run into cultural works illustrating, or inspired by the Bible. If we want to understand the cultural work, we need to understand the source material.
Mr. Gunter again:
... the Bible’s influence spreads beyond the literary realm into the artistic and the cultural. Any student of art or music will deal extensively with religious material. Moreover, biblical allusions in culture persist into the 21st century: in movie titles, song lyrics, newspaper headlines, billboards, and so forth—even television’s “The Simpsons” draws extensively from the Bible. In short, biblical knowledge enriches our understanding of both high art and popular culture.
The acrimonious debate between “creationists” and “evolutionists” would evaporate if both camps actually understood what the Bible says.
As Mr. Gunter concludes:
It is a sensitive endeavor, to be sure. But we first must recognize the value of undertaking that task. The Bible is a remarkable document, parts of which can stand with Plato in their philosophical depth, with Tolstoy in their political complexities, and with Shakespeare in their poetic beauty. The religious sphere does not have exclusive ownership over those important words. We should give our young people the tools to understand the Bible, both for their own enlightenment and to better inform their decisionmaking as citizens.