Which is the correct verb?

1. One quarter of the pizza was eaten.
2. One quarter of the class are girls.

It depends on whether one quarter is regarded as a measure of amount or of number. In (1), the emphasis is likely to be on the amount of pizza eaten, and not on the number of individual quarters, so the singular 'is' would be appropriate. In contrast, in (2) the emphasis is on the number of members of the class who are girls, so the plural 'are' is required.

Con I really gotta give ya credit

Thanks, Contrex.
1. Three quarters of the pizza was eaten. [Is was also correct with change in the fraction?]

Yes.

Three quarters (or three-quarters) of the whole pizza, considered as an amount, was eaten, but if the pizza had been first cut into four equally sized pieces ("quarters") then (of course) you would say that three of these were eaten.

I got a call at work to bring home some food. I asked why and was told,
"There's not much left. Three-quarters of the pizza is eaten"

Yes, perfect grammar ("is eaten"), although younger folk don't seem to use that form much.