Both women, each aged 53, were stabbed to death.
The 22-year-old, who had been dating Ms Graham's daughter, Shannon, told a court in Izmir, west Turkey, he was "full of regret".
"I was young and naive. I was in love," he said. "I did not plan this in advance as the prosecution claim.
"I committed the murders under severe provocation. I did it so I would not lose my girlfriend.
"I only ask for a fair trial and hope God will forgive me."
Ms Graham and her friend Ms Dinsmore, both from Newry, Co Down, were said to have tried to warn Recep Cetin away from Shannon, who was 15 at the time.
The bodies of the women were found in woods on the outskirts of Izmir in west Turkey in 2011.
They both suffered multiple stab wounds.
At Izmir Bayrakli Fifth High Criminal Court yesterday, prosecutors asked for Recep Cetin and his father Eyup, who is also accused of murder, to be given life sentences in solitary confinement.
Judges also ruled against conducting another mental health assessment on Recep Cetin, which had been requested by his defence at the last hearing.
Recep's father, Eyup Cetin, who is also accused of murder, has claimed his innocence.
The 43-year-old repeated earlier claims that secret witness evidence placing him at the scene of the crime was contradictory.
He added: "I am innocent but I have been in prison for two years. My command of Turkish is poor so I can't defend myself adequately.
"I accept any sentence the court will give if there is any evidence showing my involvement in this incident."
Relatives of the victims did not attend today's hearing.
The trial was adjourned to October 2, when a verdict is expected.