Agca's lawyers said earlier that the military authorities still considered him to be a draft dodger and required him to undergo the examination.
'Full of mistakes'
There have been long-standing questions about the 52-year-old's mental health based on his frequent outbursts and claims that he was the messiah.
Following his release from prison, Agca issued a handwritten statement through his lawyers in which he claimed the world would end "this century".
"I proclaim the end of the world. All the world will be destroyed in this century. Every human being will die in this century," he said in his statement.
"The Gospel is full of mistakes," he said, apparently referring to the Christian religious texts. "I will write the perfect Gospel."
Mehmet Ali Agca was pardoned 11 years ago in Italy and then extradited to Turkey
He signed the paper as "The Christ eternal, Mehmet Ali Agca".
Agca served 19 years in an Italian prison for his attack on the pope, before being pardoned on the pontiff's initiative in 2000.
But he was extradited to serve a sentence in his home country for other crimes, including the 1979 murder of a newspaper editor.
The pope, who died in 2005, met with Agca in Italy's Rebibbia prison in 1983 and forgave him for the attack.
Agca's motives for shooting and wounding the pope remain a mystery.
Some people believe he was working for Soviet-era eastern European security services alarmed by the Polish pontiff's fierce opposition to communism.
In a statement issued last week, Agca said he would answer questions on the attack in the next few weeks, including whether the Soviet and Bulgarian governments were involved.