I have a Peavey 8600 amplifier. It sounds great but when only operating vocals, in order to get the highs of your voice just right, the speakers put off a large amount of white noise. While singing, you can't hear it of course, but once you've stopped singing, you can hear it quite prevalently. Can this be eliminated?

If you are cranking up the highs on the equalizer(s) you can expect to get hiss at high gain levels. This is due to statistical noise in electronics. Balancing the gain levels through the whole system is important. The master gains should be around 75% up, as well as the faders for each channel. The trim (input gain) for each channel should be set so the channel doesn't go into clipping with maximum sound into the mics.
Use of speakers that do not handle vocal range well can lure one into boosting the highs to the point of raising the high frequency noise level... MOST hiss that you are hearing is NOT WHITE noise, but in the higher pitches. Most vocal frequencies are between about 250 to 4000 Hz in a typical setup.

It's quite common for this to happen. It's not white noise, it's just Hiss of all the components within the amp.
Quite common with lower end kit at high volume.

There is not enough info here to provide a complete answer, but let me give you some background:

There will always be some hiss. This is due to general amplification where a small amount of noise is amplified along with the general signal. In most cases, the signal is strong enough to overpower any noise that is present (vocals in your example). This noise should not be that noticable in normal cases. If there is an automatic gain control in the line, this could account for it as with "silent" times, the gain will automatically be increased potentially to the degree where the noise is noticable.

Barring that scenerio, if the input signal is too low, the overall gain necessary to produce reasonable volume at the speakers will also be such that the noise is noticable. In order to track this down, please provide details about the current setup including:

1) Microphone make and model being used
2) Input being used on the mixer
3) Gain level being used for the mic channel
4) Gain level being used at the master level.

Dan