Bond prices fell. The yield on the 10-year Treasury note rose to 3 per cent.
West Texas Intermediate, the US benchmark for the price of oil, was up 2.45 percent to $70.75 per barrel.
Riyadh's statement comes after Iran, earlier in the day, revealed that it had devised a plan to respond to the USA exit from the deal, without specifying the details.
The U.S. president has already said that oil prices are "artificially very high" and this "will not be accepted", while Saudi Arabia has been trying to push prices higher, and resisting calls for it to ease production curbs.
Analysts had little hope that opposition to the U.S. action would prevent sanctions from going ahead.
"Europe and China will not fight against the USA sanctions". They will grumble and accept it.
US drillers added rigs for the sixth straight week, bringing the total rig count to 844, highest since March 2015, General Electric Co's Baker Hughes energy services firm said.
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The EIA on Tuesday raised its forecast for United States output in its monthly report to 12 million bpd late next year. "This includes actions under both our primary and secondary sanctions authorities", Mnuchin added.
Now that it's left the accord, the USA intends to reimplement sanctions on Iran following certain "90-day and 180-day wind-down periods", according to Treasury Secretary Mnuchin.
"But the impact will escalate as we approach November (i.e. the end of 180-day wind down period)".
"The amount of oil that would be removed would be lower, and at this point, maybe 300,000 to 500,000 barrels a day", Daniel Yergin, vice chair of IHS Markit, told CNBC.
Oil prices have been rising in the past few weeks due to worries over the possibility that the USA could reimpose sanctions on Iran, which would then hinder Tehran's ability to export oil and create disruption in the global supply.
Futures briefly spiked to new highs after the U.S Energy Information Administration reported US stockpiles of both crude oil and gasoline fell by 2.2 million barrels in the week through May 4.
One factor that could partially mitigate any shortfall from Iran is soaring US oil output.
Outside OPEC, soaring USA crude oil production may also help fill Iran's supply gap, hitting another record last week by climbing to 10.7 million barrels per day (bpd).