he "vowed to" never do it again?
he vowed to never do it again.
he vowed never to do it again.
he vowed he would never do it again.

which of these is correct?

Question

he "vowed to" never do it again?
he vowed to never do it again.
he vowed never to do it again.
he vowed he would never do it again.

which of these is correct?

Answer

In strict English, you are not supposed to split an infinitive [ to + ].
This means that "he vowed never to do it again" is more correct than the first sentence.
The third sentence is also correct and could also be stated
"He vowed that he would never do it again."
If you were speaking, all four ways would be understood by a native English speaker.