Palmer, 47, the second person to be executed this year in Ohio, died by lethal injection at 10:35 a.m.
He said in an interview last week he deserved to be executed for killing Charles Sponhaltz and Steven Vargo in what he called his "postal moment," The Columbus Dispatch Reported.
"I killed two people. I've always accepted responsibility for the taking of their lives," Palmer said. "I believe in justice and I believe that the victims, their hatred, their anger, they need to have justice."
Before he was put to death Thursday -- in the presence of the widows of his victims, along with two of Sponhaltz's daughters, a niece and a friend of Vargo's -- Palmer said: "I want you to know I've carried you in my heart for years now. I am so sorry. I wish this could bring him back to you, but it can't. I only hope you let the pain and hurt die with me today."
Tiffany Nameth, who was married to Sponhaltz, said the apology "should have come a long time ago," the newspaper said.
"When you murder somebody what good does an apology do?" she said.
Vargo's window, Valerie Vargo-Jolliffe, said, "I'm glad this is finally over and justice is served for my late husband."
Palmer said he had been drinking and using drugs, and had been firing his gun out the window of a friend's car before the collision. He said he went to the pickup truck and, without provocation, shot Sponhaltz twice in the head -- and then shot Vargo shortly afterward when Vargo pulled over on the road.
"I didn't kill him because I thought he was a witness. I killed him because he was there," Palmer said.
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