Saudi Arabia declared diplomatic war on neighboring Gulf state Qatar this year and openly supported, and appeared to control, Lebanese Prime Minister Saad Hariri against his country's Iran-backed Hezbollah movement, which is part of the government.
He also noted "the fact that U.S. support has helped create Yemen's horrific crisis" and called on Washington to cease military assistance if parties don't agree an immediate ceasefire and a political settlement.
Tillerson criticised Riyadh's handling of conflicts in Yemen and relations with Lebanon and Qatar, and repeated U.S. President Donald Trump's call for Riyadh to end its blockade of Yemen.
As the war drags on, seven million people are believed to be on the brink of starvation and a cholera outbreak has caused more than 2,000 deaths.
"The Trump administration must continue to make clear to Saudi Arabia that the USA will not support a campaign that intentionally starves civilians into submission".
But Trump said he had asked aides to the Saudi leadership "to request that they completely allow food, fuel, water, and medicine to reach the Yemeni people who desperately need it".More news: Palestinian leader won't meet Pence in snub over Jerusalem move
Oxfam International applauded Trump's statement, calling it "long overdue but hugely important".
The Islamic Republic would "have no problem" with Saudi Arabia if it stops "bowing to Israel" and relies on itself, Rouhani said at a parliamental meeting.
Both critics and supporters of Trump say his actions to call out Saudi Arabia are welcomed news.
"We want Saudi Arabia to stop two things, the misguided friendship with Israel and the inhuman bombardment of Yemen", Rouhani said on Sunday speaking before the majlis in Tehran. Many in the region believed he resigned under Saudi pressure.
Trump's statement came after the second consecutive night of Saudi airstrikes in Yemen's capital, Sanaa, as the Saudi-led coalition tried to prevent the Houthis from consolidating power in the city.
US Defense Secretary Jim Mattis said on Tuesday that the killing of Saleh would likely worsen an already dire humanitarian situation in the country in the short term.