We present scanning tunneling microscopy and spectroscopy experiments on the novel J_eff = 1/2 Mott insulator Sr2IrO4. Local density of states (LDOS) measurements show an intrinsic insulating gap of 620 meV that is asymmetric about the Fermi level and is larger than previously reported values. The size of this gap suggests that Sr2IrO4 is likely a Mott rather than Slater insulator. In addition, we found a small number of native defects which create in-gap spectral weight. Atomically resolved LDOS measurements on and off the defects shows that this energy gap is quite fragile. Together the extended nature of the 5d electrons and poor screening of defects help explain the elusive nature of this gap.