Ungrammatical: "Half the boys jumped, but only a quarter of the girls did."?

The sentence is fine grammatically, and the pattern is very common in English in both informal and formal speech and writing.

Did is an auxiliary verb, but what is more interesting is what is NOT there: the verb jump is understood to be there grammatically, just like you is understood to be the subject in Stop criticizing my grammar!

This is an example of verb phrase ellipsis.

From Wikipedia:

Verb phrase ellipsis

Verb phrase ellipsis (also VP-ellipsis or VPE) is a particularly frequent form of ellipsis in English. VP-ellipsis elides a non-finite VP. The ellipsis must be introduced by an auxiliary verb or by the particle to.

John can play the guitar; Mary can [play the guitar], too. - VP-ellipsis
He has done it before, which means he will [do it] again. - VP-ellipsis

An aspect of VP-ellipsis that is unlike gapping and stripping is that it can occur forwards or backwards. That is, the ellipsis can precede or follow its antecedent, e.g.

The man who wanted to [order the salmon] did order the salmon. - VP-ellipsis
The man who wanted to order the salmon did [order the salmon]. - VP-ellipsis