Ignoramus25553 wrote:

(...)

In other words, is the voltage drop on a 1,000V rated diode, the same
as on a 30v rated diode?

Yup. Forward voltage is largely independent of
Peak Reverse Voltage.
Here is a part available in PIVs from
50 V to 1 KV:
http://www.fairchildsemi.com/ds/1N/1N4007.pdf

Notice that Vf at 1A is the same for all flavors
at 1.1 V in 'non-pulse' service,
largely due to the material spec of each
side of the junction.

Like all good rules, this one has an exception, for
'rectifiers' made up of series diode elements to
withstand very high peak inverse voltages will have
one diode drop for each diode in the string.
This device for example:
http://www.hvpsi.com/pdf/D4468A01r03.pdf

...has 22 diodes in series to provide rectification
at 220 KV.
We can expect forward voltage to be upwards of
17-24 V at maximum current for this device because
we have to saturate all 22 junctions before the
device turns on.

--Winston