Al-Hakim, 35, was taken into custody at the Zirbatyah crossing point southeast of Baghdad along with his security guards, said his father's secretary, Jamal al-Sagheer.
Zalmay Khalilzad, the US ambassador to Iraq, issued an apology amid concerns that the detention would prompt a backlash from Shia leaders.
The US military called the incident "unfortunate", and said the vehicles were stopped because they "met specific criteria for further investigation in an area where smuggling activity has taken place in the past."
Al-Hakim was detained after members of the convoy "did not co-operate with coalition forces and displayed suspicious activities," the US military said.
They said he was released to Iraqi authorities and his possessions were returned after further investigation.
Also on Saturday, a suicide car bomber attacked a checkpoint protecting Abdel Aziz Hakim’s home, killing at least three people and wounding seven.
An interior ministry official said the attacker struck the first barrier protecting an access road leading to Hakim's fortified compound in downtown Baghdad's Jadriyah district.
"He tried to break through the first barrier towards Hakim's house. The guards opened fire on him and he blew himself up," said the official, speaking on condition of anonymity.
In a separate incident, gunmen in two vehicles stormed an Iraqi police checkpoint near Baghdad airport, killing eight police officers and wounding two, the US military said.
Two of the attackers were killed when the US military arrived to support the police.