The Yemeni rebel group used a cruise missile to strike an oil site near the city of Jeddah on the Red Sea coast.
The energy secretary said Iran "could pay a very dear price economically" but added that Trump is "not interested in going to war."
Russian Foreign Ministry spokeswoman Maria Zakharova also suggested the U.S. strategy in the Middle East is collapsing.
The White House has been considering how to respond to the weekend's attacks on Saudi oil infrastructure, which both Washington and Riyadh are blaming on Iran.
Retired Admiral William McRaven said the president's team must draw on their expertise to provide Trump with a range of options to respond to the attacks on Saudi Arabia.
The Houthis have claimed responsibility for Saturday's attack, but the U.S. has suggested that Iran is ultimately to blame.
"What if this whole thing goes south?" Kennedy asked during a segment of Fox News Outnumbered on Monday.
The Democratic presidential candidate criticized the president after he said he was "waiting to hear" from Saudi Arabia before acting in regards to the weekend Aramco drone attack.
"There's a vacuum of leadership inside the White House," the former White House communications director said.
"This is why it's so important to have a president who isn't a typical politician," the senior White House counselor said.
"Such [American] comments and measures are more akin to the plots hatched by secret and intelligence services for damaging the image of a state to prepare the ground for a series of [hostile] measures," the Iranian Foreign Ministry said.
In Yemen and beyond, these inhumane weapons serve virtually no military purpose and kill indiscriminately.
"More than 10,000 displaced households (nearly 80,000 people) have been assisted with food, emergency kits and other life-saving support."
Despite reports that the Saudi-led coalition had seized a crucial port city, a top Iranian official said the Houthis were "witnessing their advances every day."
"The U.S. must stop backing this war and support the UN's effort to get a cease-fire."
Hodeidah is a vital access point for humanitarian and military aid for the Houthi rebels.
The spread of drone technology has allowed militant groups to build de facto air forces.
Local officials said jets continued flying over the site after the attack, keeping rescuers from reaching the victims.
On Wednesday, Saudi Arabia said it had intercepted another three Houthi ballistic missiles fired at its cities.
A Saudi royal decree has forced the kingdom's chief of the joint staff and air force commander out of their positions.
Will U.N. ambassador Nikki Haley, like former Secretary of State Colin Powell, regret her presentation on the alleged threat posed by Iran?
A new report showed Yemen's Iran-friendly Houthi movement evaded the U.S. anti-missile system protecting troops from North Korea.
The administration's bellicose rhetoric might be hurting Kusher's diplomatic efforts.
Footage on social media appeared to show the former leader's body wrapped in a blanket and a large wound on the side of his head.
Iran has rallied groups across the Middle East against what it perceives to be the combined interests of ISIS, Israel, Saudi Arabia and the U.S.