"ISIS has a new leader. We know exactly who he is!" Trump tweeted, less than a week after a US-led commando raid killed the group's leader Abu Bakr al-Baghdadi.
Islamic State on Thursday confirmed Baghdadi's death and named his replacement as Abu Ibrahim al-Hashimi al-Quraishi.
However, little is known about Hashimi, whose name was seldom mentioned as a possible successor the multiple times that Baghdadi was falsely reported killed in recent years.
Nathan Sales, the State Department counterterrorism coordinator, provided no details on Hashimi when asked by reporters, saying only that the United States was "looking into the leader, his role, the organization where he came from."
"We will continue to subject that organization to unrelenting counterterrorism pressure using all the tools of national power," Sales said.
"We will dismantle the group regardless of who its leadership cadre is," he said.
Baghdadi, who led IS since 2014 and was the world's most wanted man, died in a US special forces raid in Syria's northwestern province of Idlib on Sunday.
Islamic State also confirmed the killing in another raid the following day of the group's previous spokesman, Abu Hassan al-Muhajir.
The statement said the jihadist group's legislative and consultative body convened after the 48-year-old Iraqi-born jihadist chief's death and "agreed" on a replacement.
An IS spokesman denounced Trump as a "crazy old man" and warned that the group would avenge Baghdadi's death.
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