Maddon argued that pinch-runner Rich Thompson, who was attempting to steal second base, was safe, but Davidson ruled that shortstop Derek Jeter applied a tag before Thompson reached the bag.
"Retrospectively, I've heard Bob was right," Maddon said. "Bob, I'm telling you, you were right. I apologize. I didn't say anything really bad. I really enjoy Bob. Bob is an old-school umpire, and he and I have done this before."
Three batters after Maddon was ejected, Chris Gimenez snapped a 3-3 tie with a two-out RBI single that scored Ryan Roberts, who had stolen second base after singling.
The ejection was the third of the season for Maddon and his 26th as the manager of the Rays.
"He did not want to throw me out," Maddon joked. "He said some funny things, and I had to keep a straight face and keep arguing in spite of it because he's a very entertaining umpire on top of that."