Turkey on Sunday said there was "coordination" between Ankara and Washington before the operation which USA media reports said targeted and killed Islamic State chief Abu Bakr al-Baghdadi.
McKenzie did not directly corroborate Trump's statement, which he made during wide-ranging remarks on Sunday while announcing the operation, that Baghdadi was "whimpering and crying" during his final moments.
He said two children, and not three as initially reported, died when Baghdadi blew himself up.
Al-Mujahir was killed on Sunday in an airstrike elsewhere in northern Syria.
The Pentagon released the first government photos and video clips of the nighttime operation, including one showing Delta Force commandos approaching the walls of the compound in which Al Baghdadi and others were found.
The new spokesman, named Abu Hamza al-Qurayshi, urged followers to pledge allegiance to the new Caliph and addressed the Americans, saying: "Don't rejoice".
The name al-Qurashi appears to identify Hashemi as a descended of the Prophet Muhammad's Quraysh tribe - which is generally considered a qualification for a becoming a caliph and was also used by Baghdadi in his full title.
But Mr Morrison said this was not the end of ISIS, warning the group's "perverted ideology" continued to pose a national security threat.
"America, don't you realize that the Islamic State is now at the forefront of Europe and West Africa?" the group - actually running on fumes for some time, per most intelligence reports - taunted in its message, according to The Sun.
He stole a pair of al-Baghdadi's underpants to prove his connection to the terror leader and then led a U.S. special forces team to the Syrian hideout.
Among the images the Defense Department released was grainy black-and-white footage of U.S. troops approaching the high-walled compound on foot, as well as video that showed about a dozen unidentified fighters on the ground opening fire on the helicopters that were ferrying the United States troops to the site.
"The Islamic State shura council convened immediately after confirming the martyrdom of Sheikh Abu Bakr al-Baghdadi, and the elders of the holy warriors agreed" on a replacement, said the seven-minute message.
Islamic State additionally confirmed the dying of its spokesman Abu al-Hassan al-Muhajir.
Fighters from the newly-formed IS group that year swept through much of the Sunni heartland in Iraq and Syria to declare a "caliphate" that further expanded to reach roughly the size of Great Britain.