Iran's Revolutionary Guards said they reserved the right to retaliate against any ceasefire violations, while Iran's Supreme Leader Mojtaba Khamenei said countries in the region could no longer be a shield for US bases.
Both sides had indicated progress on a memorandum of understanding that could halt the war and restart shipping through the blockaded Strait of Hormuz, while giving negotiators 60 days to negotiate more complex issues including Iran's nuclear program.
Referring to the strikes against targets including boats attempting to lay mines and missile launch sites, Rubio told reporters on his plane in India's Jaipur that the Strait of Hormuz has to be open "one way or the other".
The war, which began with US and Israeli strikes on Iran on February 28, has caused an unprecedented oil supply shock, pushing up oil prices along with the costs of fuel, fertiliser and food.
Iran responded to the strikes by launching drones and missiles at Gulf states that host US bases.
Only a few dozen vessels have been passing through the Strait of Hormuz compared with 125 to 140 daily previously.
About a fifth of the world's oil and liquefied natural gas usually flows through the waterway.
Iran has been letting some ships through, giving preference to ships linked to countries with which it has alliances or close ties, and striking government-to-government agreements, Reuters has reported.
Despite a ceasefire in place since early April, US Central Command said in a statement on Monday it had carried out fresh strikes designed "to protect our troops from threats posed by Iranian forces."
Iran said on Monday it had downed a "hostile" stealth drone using a new air defence system, without saying where it had come from.
The Revolutionary Guards said air defence units had also fired at another drone and at a fighter jet which they said had entered Iranian airspace over the Gulf region.
In comments posted on his Telegram channel on the occasion of the annual hajj pilgrimage, Iran's Supreme Leader said: "The clock cannot be turned back, and the nations and lands of the region will no longer be a shield for American bases."
"From now on, the slogans 'Death to America' and 'Death to Israel' will be the slogans of the Islamic nation and the oppressed people of the world, especially the youth," he added.
US President Donald Trump has previously cited the slogans while justifying military action against Iran.
Trump had said talks with Iran were going "nicely" in a lengthy post on Truth Social on Monday, but warned of fresh attacks if they failed. It "will only be a Great Deal for all, or no Deal at all," he wrote.
Trump also called on more Arab and Muslim states including Saudi Arabia to sign up to the Abraham Accords, brokered during his first term in office and aimed at normalising ties between those states and Israel.
Saudi Arabia's longstanding position has been that it would not sign the accords unless there is an agreement on a roadmap to Palestinian statehood.
In another indication of the region's tensions, Israeli Prime Minister Benjamin Netanyahu said on Monday Israel would intensify strikes against the Iran-backed Hezbollah militia in Lebanon.
Israel's military soon thereafter said it was attacking Hezbollah infrastructure in Lebanon's eastern Bekaa Valley and other areas.
Israel and Lebanon agreed to a ceasefire in mid-April, but Israel has continued airstrikes it says are acts of self-defence against Hezbollah, which was not party to the truce.
