Does Having the Turntable Grounded Through the Right RCA Effect Sound Quality?

No, ground is ground. It is either grounded or it is not. The extra ground wire or terminal is doing the same thing, grounding. The only detriment to not grounding is hum or buzz....if it is not doing that, then you are properly grounded. Ground does not carry a signal. 

@cleeds, yes. You understood what I was asking. I was specifically asking about certain turntables that have a grounding scheme that uses the negative on the right channel on the RCA connection, rather than through a terminal on the preamp. Thorens, Dual and some Rega  models (maybe?) are like this. I had read some place that changing from this scheme to the more common one can have positive results on the clarity and quality.

Sorry if I wasn't clear with how I worded it.

Now that that's cleared up, does anyone have any thoughts on it?

One difference from one installation to another is whether the audio ground is identical to chassis or AC ground.  Some designers like to place a small value resistance between audio and chassis ground, for noise abatement.  Some use an arrangement of diodes to do that, also. So, if your phono stage has a direct connection between audio and chassis ground, then apart from the freakishness of grounds in general, I don't see a reason why having a separate wire to a lug on the chassis vs incorporating it into one lead or the other audio ground would make much difference, except maybe where the quality of the connection per se might make a difference.