Accepting that LvW has the complete answer, I'd just like to try to help you a bit with understanding.
needhelp171 said:
... Once the diode begins conducting current, the voltages at the anode and the cathode are identical (the diode serves as an open wire).
That's wrong - on both counts.
Would you say, once a resistor starts conducting the voltages at each end are the same? (I hope not.)
Diodes do not behave like a piece of wire (if you mean by that, a zero resistance conductor, which of course normal wires are not.) You can talk to LvW about how they do behave, but i hope he might not mind if I say that the PD across the diode does not decease when it conducts. As with a resistor (albeit in a different mathematical way) the PD increases with current.
needhelp171 said:
I heard that the voltage at the anode must be at least 0.7 volts higher than the voltage at the cathode. Is that true, even for an ideal diode? I know the constant 0.7 may differ, but I'm just looking for a general principle.
I'd have said, you can say what you like for an ideal diode - just depends on what you ideally want it to do. But mostly when we talk about an ideal diode there is no forward voltage drop nor resistance: it justs conducts in the forward direction and does not conduct in the reverse direction.
Since no real diodes behave like that, we may use other ideal models. I think that is what you have in mind: it does not conduct up to 0.7V, then it conducts with no further increase in PD, but no drop in PD either - just fixed at 0.7V.
Real diodes don't do that either. It is just a different ideal model.
Some real diodes conduct something like LvW's Shockley equation (maybe all afaik, I'm just hedging my bets because I know LvW of old!) That means (if I understand it correctly - just looking at the equation as a mathematical expression) that the current is exponentially related to the PD when it is forward biased, which in turn would mean (and I'm getting on very soft ground here) that it conducts for all PDs except 0. But for small PDs the current it conducts is tiny. Then around this magic voltage 0.7V (or whatever it is, as you say) over a relatively small change in PD (a few 100 mV) the current increases drasticaly from microamps to amps. At that point you can get large changes in current for small changes in voltage as if you had a small resistor in series with a 0.7V source.
(Purely a personal opinion: I think a Si diode is conducting enough for me to take an interest around 0.5V and could be 1.0V by the time it is conducting amps. But I still accept the 0.7 model most of the time.)
Your general principle is correct - you never get rid of the 0.7V (or whichever). Below that it is a poor or non conductor. Above it conducts well with a small resistance and the 0.7V drop.
I'm not disagreeing with anyone about reading about the Shockley equation. But I would say that for me, it is helpful to look at a graph of current against PD for a diode (preferably a real one if you can find one, but if not, even an idealised or calculated one will give you an idea of what diodes actually do.)
