Tensions have soared since the U.S. airstrike earlier this month that killed Gen. Qassem Soleimani.
President Rouhani warned that the presence of foreign forces could cause problems for the world's "energy security."
Iran has begun using an array of 20 IR-6 centrifuges.
The U.S. Navy is trying to put together a new coalition of nations to counter what it sees as a maritime threat from Iran.
Members of the 11th Marine Expeditionary Unit were photographed this week with a Light Armored Vehicle on the flight deck.
Two Royal Navy warships are currently in the region, the frigate HMS Montrose and the destroyer HMS Duncan.
This marks Iran's third seizure of a commercial vessel in recent weeks.
The U.S. military's main role would be to provide "maritime domain awareness" to coalition partners in vulnerable waterways.
Officials with CENTCOM say the U.S. was never "warned" before Iran took out the drone, contrary to the regime's statements.
A senior official said the entire crew of the Stena Impero oil tanker was in good health.