Basem Ali Mohammed al-Gidaihi "died yesterday (Saturday) afternoon," ministry spokesman General Mansur al-Turki told AFP.
Gidaihi was wanted on charges including previously shooting at police and taking part in "riots," SPA reported citing the interior ministry.
Gidaihi was taken to hospital after the exchange of fire in the village of Awamiya, the scene of past clashes with protesters, but suffered a "setback" following initial treatment, the official Saudi Press Agency said, citing doctors.
An associate of the suspect was in stable condition after being wounded in the gunfight, the doctors added.
Awamiya is just west of Dammam city in oil-rich Eastern Province.
Demonstrations in Eastern Province erupted in 2011 alongside a protest movement in neighbouring Bahrain.
They turned violent in 2012 and clashes between police and protesters killed at least 24 people, including at least four policemen.
In June, a court sentenced two people to death for "taking part in forming a terrorist group" and other crimes linked to the protests.
Saudi Shias complain of widespread discrimination, which they say prevents them from gaining good government jobs, reduces state investment in their neighbourhoods and leads to closures of Shia centres of worship.
The Saudi government denies charges of discrimination.
The last official Saudi figures in 2001 showed there were fewer than one million Shias in Eastern Province out of a total 3 million people.
However, a 2006 Human Rights Watch report put the number of Shias there at 2 million and a 2008 US diplomatic cable released by WikiLeaks estimated the Shias at 1.5 million
(AFP, Reuters)