If I want to run a 175 watt mercury vapor lamp off this ballast, and power factor correction is not an issue, do I need the capacitor? I would think not since the lamp in question has a starting electrode. Is this correct?

I'm almost certain you still need the capacitor. It's got something to do with the inductor-capacitor reactance or something like that. I'm sure someone else will elaborate.

Generally, if the capacitor is specified as optional, you do not need it.

Otherwise it depend on ballast type:
CWA or mag-reg (cap in series with secondary or on separate winding): It should be there, should have exactly the specified value, tolerance same as specified or better and higher or equal voltage rating.

On HX it depend, where the capacitor should be connected. If directly to the mains input, you might omit it, as it only correct the PFC and you say this is not an issue for you.

If the capacitor (on multi-voltage tapped HX ballast) is connected between some higher voltage tap and neutral (e.g. on 120/240/277V it is connected between 277 and N), it's omitting would yield to slightly higher losses (on primary section), so you have to be careful for ballast temperature during operation and better foresee some heatsinking measure.