Earthquakes are generally caused by unstable stick-slip motion of faults. This stick-slip phenomenon, along with other frictional properties of materials at the macroscale, is well-described by empirical rate and state friction (RSF) laws. Here we study stick-slip behavior for nanoscale single-asperity silica-silica contacts in atomic force microscopy experiments. The stick-slip is quasiperiodic, and both the amplitude and spatial period of stick-slip increase with normal load and decrease with the loading point (i.e., scanning) velocity. The peak force prior to each slip increases with the temporal period logarithmically, and decreases with velocity logarithmically, consistent with stick-slip behavior at the macroscale. However, unlike macroscale behavior, the minimum force after each slip is independent of velocity. The temporal period scales with velocity in a nearly power law fashion with an exponent between -1 and -2, similar to macroscale behavior. With increasing velocity, stick-slip behavior transitions into steady sliding. In the transition regime between stick-slip and smooth sliding, some slip events exhibit only partial force drops. The results are interpreted in the context of interfacial chemical bond formation and rate effects previously identified for nanoscale contacts. These results contribute to a physical picture of interfacial chemical bond-induced stick-slip, and further establish RSF laws at the nanoscale.