Tehran said any move to detain the vessel would be an error.
"Iran has given necessary warnings to American officials through its official channels...not to make such a mistake because it would have grave consequences," said foreign ministry spokesman Abbas Mousavi.
He added the American sanctions against Iran had no legal basis. "The US should come to its senses. Bullying and unilateralism cannot get anywhere in the world today,” he declared.
Tensions between Iran and the United States have been on the rise since President Donald Trump unilaterally withdrew from a landmark nuclear deal in May 2018 and began imposing sanctions against the Islamic republic.
The Grace 1, now renamed the Adrian Darya 1, departed Gibraltar late Sunday headed for Greece, according to the monitoring website Marine Traffic.
The tracking site showed the vessel moving east into the Mediterranean and listed Kalamata in Greece as the destination. The Greek authorities were yet to confirm that it is expected to dock there.
The tanker was seized off the coast of the British overseas territory of Gibraltar on 4 July on suspicion that it was carrying oil to Syria, in breach of European Union sanctions. After more than 40 days, on 15 August, Gibraltar's Supreme Court ordered the release of the vessel.
The decision came after Gibraltar's government said it had received written assurances from Iran that the ship would not be headed for countries "subject to European Union sanctions".
On Friday 16 August, a US court issued a warrant for the seizure of the tanker, on the grounds that it had links to Iran's Islamic Revolutionary Guard Corps (IRGC), which Washington has designated a "terrorist" organisation.
- Gibraltar refuses US demand to detain Iranian supertanker
- US issues warrant to seize Iranian Grace 1 tanker in Gibraltar
Gibraltar said on Sunday that it was unable to comply with the US request because it was bound by EU law and US sanctions against Iran were not applicable in the EU. The authorities also said the Revolutionary Guard was not designated a terrorist organisation under EU, UK or Gibraltar law.