If we don't reach 0.6 V across the diode, do we just approximate the current as 0?

Yes, in the simplest model. If there's less than $\sim$0.6 V (for silicon) across, there will be some depletion zone left and so poor conductivity (there will still be a little bit of current flow from minority carriers). For a diode to be fully forward-biased, there must be at least $V_{pn}$ across it.