I'm gonna say that this sentence is not correct. It should read:
"I repeatedly asked my son if he broke the window, and he swore that he did not."
The first example may sound correct but it is not.
I have pinged an experienced eye as an arbiter. Mainly because I'm only 98% certain of my claim.
May I ask your indulgence Mademoiselle?
I would contend that the oath or affirmation provides for an ongoing, present condition or state. Until relieved of the oath or the swearing is found to be invalid (i.e. determined to be a lie), his swearing continues, thus, “he swears.”
Hmmm. “He swore that he did not” means she asked, and he answered.
“He swears that he did not” sounds like he’s been asked repeatedly and his story hasn’t changed.
So the correctness would depend on which meaning is aimed at.