Curiously, however, Mr. Trump was mostly interested in settling old scores the day after his nomination became official. Clearly still stinging from Mr. Cruz’s actions, Mr. Trump ruminated aloud about why the senator from Texas would not back him. He recalled their personal fights during the nominating contests, including an unflattering picture of Mr. Cruz’s wife, Heidi, that Mr. Trump reposted on Twitter. Mr. Trump also defended his suggestion that Mr. Cruz’s father might have played a role in the assassination of President John F. Kennedy, citing a report in The National Enquirer.

“This is a magazine that, frankly, in many respects, should be very respected,” Mr. Trump said of the tabloid.

Mr. Trump’s comments about Ms. Cruz — which prompted an audible gasp from a member of his traveling press corps — is also likely to remind voters that he attacked her during the primary contest. Mr. Trump is already struggling to woo female voters, who still view him overwhelmingly negatively, polls show.

Trying to explain away the episode also left him in the slightly awkward situation of mentioning Penthouse, a pornographic magazine, while Gov. Mike Pence of Indiana, Mr. Trump’s running mate and a mild-mannered social conservative, stood looking on just behind him. (Mr. Trump mentioned the magazine in asserting that a nearly nude picture of his wife was “artsy” and in GQ magazine, which he said was hardly Penthouse.)

At one point, the Republican nominee quipped that, as president, he might form a “super PAC” to challenge a Senate re-election bid by Mr. Cruz, before questioning the ethics of such a move. And he kept up the personal attacks, saying that Mr. Cruz did not know how to properly use his intellect.

“He’s got good intellect, but he doesn’t know how to use it; and he was a good debater, but he didn’t do well in the debates against me,” Mr. Trump said.