Whatever happened that night remains unclear.
Both Griffey and manager Don Wakamatsu said the designated hitter was available for pinch-hit duty at that time, but it remains unclear whether an in-game snooze occurred at some other point during the game.
"I really don't want to go into it anymore," said Griffey on Friday night when asked if he wanted to tell his side of the story. "I want to leave it alone and get on with the season."
The season continued in the series opener with Mike Sweeney in the designated-hitter spot against Rays right-hander Wade Davis.
"We're looking at the best matchup from day-to-day," Wakamatsu said. "We'll go with what gives us the best opportunity to win the game."
Wakamatsu used Griffey in the DH role in Tuesday night's series opener against the Orioles, and he went 0-for-3 with a walk and scored a run. Jose Lopez handled the DH duties the following night, going 1-for-4, and Sweeney hit a home run and walk in four plate appearances on Thursday.
Neither Griffey (0-for-5) nor Sweeney (0-for-3) have had success against Rays right-hander James Shields, the scheduled starter Saturday afternoon.
"I don't fill out the lineup card," Griffey said. "I just wait and hopefully I'll get another chance."
Griffey has struggled this season, batting .200 (18-for-80) with no home runs and six RBIs, the most recent coming on Wednesday when he hit a sacrifice fly as a pinch-hitter in the ninth inning.
The 40-year-old believes there still is some fuel left in his tank.
"If I didn't feel that I could compete, then why would I show up?" he said. "Why would I take early batting practice? Why would I take batting practice, period? That's just the way it is.
"I'm not there [retirement] yet," he said. "I will figure that out when I get there, but I'm not there. I'm just trying to do the things I have always done, stay in the same routine, take early batting practice when it's available.
"I am still part of this team, so I have to prepare myself day in and day out. That's it. There is a lot of baseball left, and a lot of things can happen.
"But right now, I don't worry about anything but today. Tomorrow I will worry about tomorrow."
At least for the next few days, he will get to visit his family in Orlando, Fla., a 75-minute drive from Tropicana Field.
"It's good to get to see my house," he said. "There are a couple of new things there."