Houthis and Saleh loyalists jointly overran Sanaa and much of the country in 2014, forcing the internationally recognized government of President Abd Rabbuh Mansour Hadi to flee to Saudi Arabia.
The call came as his supporters battled Houthi fighters for a fourth day in the capital Sanaa as the two sides traded blame for a rift between allies that could affect the course of the civil war.
Tension between Saleh and Huthi has been rising in recent months.
The Houthi-run al-Masira television channel said Thursday's missile hit a military target inside Saudi Arabia, but the coalition contradicted that claim.
The violence erupted after Houthi rebels attempted to arrest Brigadier Tarek Saleh, a nephew of the former president, according to an Anadolu Agency reporter.
Explosions were heard in the Yemeni capital until dawn and the neighbourhoods are closed to passersby on Saturday. There was no immediate word on casualties.
According to Saleh, the letter outlines what King Faisal described as Egypt's unsafe role in Yemen and the region in general through its support for the rebels and provoking the people's emotions against "us all", i.e. the U.S. and Saudi Arabia, as well as Egypt's ongoing and historical efforts to overthrow "our governments".
The fighting began on Wednesday when Saleh's GPC party accused the Houthis of breaking into the city's main mosque complex and firing RPGs and grenades.
According to United Nations officials, more than 10,000 people have been killed in the war, while more than 11 percent of the country's population has been displaced as a direct result of the conflict.