It was not about glory on the baseball field. Instead, they were two combat pilots talking.
Williams, a veteran of World War II and Korea, was telling the veteran of Vietnam about when his Marine jet was hit over North Korea and he chose to crash-land rather than eject and risk breaking his legs. He wanted, above all, to be able to play baseball again.
"The most fascinating few hours of my life," McCain says of that conversation. The GOP presidential contender told the story when The Associated Press asked him and other candidates to name a prized possession.
His answer: an autographed baseball from the Red Sox great, "my childhood hero, a Marine pilot as well."
In a striking coincidence, it turns out that treasure is not unique to McCain in the field of presidential hopefuls. When the AP asked other candidates the same question about their keepsakes, Democratsaid the same thing.
Richardson was 6 or 7 when his dad took him to Fenway Park to see the Red Sox play. Like countless kids pressing toward stadium titans, he came face to face with Williams and asked him to sign a ball. Williams did.
Richardson, now New Mexico governor, grew into a talented prep school and college pitcher with professional prospects before his life took other turns.
McCain visited Williams at his Florida home in the summer of 1997, when the towering and moody Hall of Famer was confined to a wheelchair. Esquire magazine arranged the meeting after McCain told editors he had admired Williams since he was little.
Williams went off to World War II after winning the American League's Triple Crown - highest batting average, most homers, most runs batted in - in 1942. He saw no combat. But the reserves called him to active duty in 1952 and he flew nearly 40 combat missions in Korea, not pleased to have his career interrupted again, but resigned to his duty.
In many missions, he was wingman for John Glenn, who would become the first American to orbit Earth and then a Democratic senator from Ohio.
"John Glenn said he was the best natural pilot he ever flew with," McCain recalled. One day, McCain said, "Ted Williams was flying a combat mission, his plane gets shot up. His plane is on fire ... and incredibly he lands wheels up at an air base, one of the most incredible feats of aviation skill.
"I said, 'Why didn't you eject?"'
Williams risked his life for his limbs. He told McCain about a metal bow stretched across the canopy, a barrier he figured his long frame would never clear. "He said, 'And I looked at my knees and I knew that I was going to break both my knees if I ejected and never play baseball again,"' McCain recounted.
Surely few could appreciate that comment more than McCain.
In October 1967, McCain broke his arms and a knee when he ejected from his stricken Navy jet on his 23rd bombing mission in Vietnam, parachuting unconscious into a Hanoi lake. Barely surviving that ordeal, he then endured torture during five years as a POW.
In February 1953, Glenn was in the air when he saw Williams bring his burning F-9F Panther down. It skidded along the runway on its belly and stopped. "I looked over my shoulder and saw this big, tall figure scrambling out of that cockpit," he wrote. "I never saw a guy move that fast in all my life."
Sportswriter Michael Barber witnessed the 1997 meeting in Williams' home and recalled the old man being wheeled into the room, roaring, "Where is the next president of the United States?" At the time, McCain's first presidential run was more than a gleam in his eye but decidedly formative.
The two swapped war stories, letting their lunch go cold, Barber wrote in the Bradenton (Fla.) Herald on the occasion of Williams' death five years later.
"McCain was just another awestruck American, marveling in the presence of a man whose splendid ability to excel at some of life's most challenging pursuits amazed us all," Barber wrote.
McCain had watched Williams at Griffith Park, home of the old Washington Senators, in his youth, and wanted to talk baseball as well as aviation. But the old man wanted to know about McCain's time in Vietnam. He autographed a baseball, describing the senator as a hero in the inscription. They became friends after that.
Yet when McCain ran for the GOP presidential nomination in 2000, Williams endorsed George W. Bush.
That seems not to have made McCain's baseball any less special.
"I will never forget that story that he told me," McCain told the AP. "Yeah, so, baseball signed by Ted Williams, I think, is probably one of my most treasured possessions."