The only text in the Bible that could conceivably be used to demonstrate that slavery is unchristian is Paul's letter to Philemon, which he sent back to Philemon along with the escaped slave Onesimus. In that letter he pleaded with Philemon to treat Onesimus with fairness "as a brother in Christ" or to perhaps even consider freeing him.

It could be that Paul simply didn't have the guts to tell people to stop owning slaves. It's far more likely that he didn't see anything wrong with slavery per se, just the mistreatment of slaves.