Swindal was arrested early Thursday by the St. Petersburg Police, who claimed he was driving 61 mph in a 35 mph zone. Swindal was released on $250 bond and has a March 15 court date.
"I just wanted to tell you on a personal level how embarrassed I am for my family, for the Yankees, the fans. I know I let a lot of people down, and I promise to do better and put this behind me," he said, standing in front of a photograph of Billy Martin in the Legends Field press box.
Steinbrenner said in June 2005 that Swindal eventually would succeed him as the Yankees' controlling owner.
Asked whether his status with the team will change, Swindal responded: `Hopefully, not." Pressed whether he still viewed himself as Steinbrenner's successor, Swindal replied: "I can't answer that other than it would be speculation."
The 52-year-old Swindal, who lives in Tampa, is a general partner of the team and is married to Steinbrenner's daughter, Jennifer. Swindal said he would be at spring training "for the duration."
Swindal said he had been instructed not to comment on the details of what led to his arrest. He talked generally about his relationship with Steinbrenner.
"I've been married 23 years and guys, I've got to tell you, on a personal level, he's been a great father-in-law. That's not changed," he said. "He's been a great father-in-law and supportive. He supports me. I don't feel anything but a guy who is looking out after me and supports me."
He said Steinbrenner's advice to him was: "Let's just try to get this behind us and move forward and get to work."