In a tweet Saturday, the president said he's now suing "various people" for confidentiality agreement violations.
Mr Trump a year ago unilaterally withdrew from the landmark 2015 global deal that placed limits on Iran's nuclear activities in exchange for the lifting of sanctions, and he reimposed crippling financial penalties.
But Mr Trump tweeted a high-resolution picture of the location, with annotations pointing to damaged vehicles and the launch gantry, saying it involved Iran's Safir satellite rocket. To drive that point home, Trump took to Twitter and released what appears to be a USA reconnaissance satellite photo of Iran's launch site after the failed launch.
The Nahid-1 is believed to be the satellite Iran was prepping to launch at the space center, which is some 240 kilometers (150 miles) southeast of Iran's capital, Tehran.
Facing media criticism for publishing a presumably "classified" image, the president resolved the question hours later on Friday night on his way to Camp David by noting, "We had a photo, and I released it, which I have the absolute right to do".
Analysts had linked the explosion to Iran's planned launch of the Nahid-1.
Asked about the explosion that led to the failed launch, Mohammad Javad Azari-Jahromi, the communications minister, said the defense ministry should announce incident details, Iran's semi-official Mehr news agency said.
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Jahromi, a rising politician in Iran's Shiite theocracy, responded to Trump in a tweet early Saturday with a selfie next to Iran's Nahid-1 at a lab in Tehran.
The Iranian president said the full lifting of all anti-Iran sanctions would lay the ground for future negotiations, stressing that if Europe fails to fulfill its commitments, Iran will implement the third phase of reductions in its commitments to the nuclear deal.
But Mr Rouhani has downplayed the likelihood of that happening unless the United States first lifts crippling sanctions that it has slapped on Iran since pulling out of the deal.
Tehran denies the United States accusation that such activity is a cover for ballistic missile development.
The Trump administration has ratcheted up economic pressure on Iran this year with economic sanctions to try to force it to renegotiate a pact reached with world powers in 2015 limiting its nuclear programme.
Last August, the Trump campaign reportedly filed for arbitration against her for violating a non-disclosure agreement she signed then violated with her book "Unhinged".
Iran has recently begun to break the accord itself while trying to push Europe to help it sell oil overseas.