Ortiz spoke at length Saturday afternoon about the incident, which occurred in the eighth inning of Boston's 10-3 victory over Baltimore Friday night.
"I don't feel proud of a situation like that, but I'm the kind of person that don't like being pushed, you know what I'm saying? I feel like I was pushed last night to do something like that," said Ortiz, who likely faces a suspension. "Everybody has a button, and once it gets touched, it changes your mind in a second. So, I guess that was my time."
Red Sox manager Terry Francona said he doesn't expect to hear from MLB until after the All-Star break on suspensions and fines.
"It's a nice time for the league, so I don't think they want to be talking about what happened last night. I wouldn't be surprised if we didn't hear anything until later," Francona said.
During Ortiz's at-bat with one out and Josh Reddick at third base, Gregg threw three straight inside fastballs and on the third one, Ortiz started walking toward the mound, pointing at the pitcher. The benches and bullpens emptied, but the situation was brought under control and play resumed.
On the ensuing pitch, Ortiz popped out to shallow center field, and when he started down the first-base line, Gregg began pointing and yelling at him. Ortiz charged Gregg and the two began throwing punches. The benches and bullpens emptied again.
"This is a guy that I always face and he never pitches in," Ortiz said. "He throws a whole bunch of pitches inside, and I'm quite sure he was trying to hit me, no question about that."
Gregg's complaint on Friday was that Ortiz swung and hit a 3-0 pitch with the Red Sox leading by seven runs, and that Boston's designated hitter didn't run it out.
"After I hit the fly ball, he started screaming at me," Ortiz said. "I ain't gonna take that like a little b----, you know what I'm saying? Everybody's a grown[up] man here, and you gotta be aware of the situation.
"I mean, you saw the argument before, and after that you're gonna act like you're my daddy? I ain't gonna take that. That ain't me, though. I haven't fought since I was in kindergarten."
Ortiz also apologized to the fans.
"That's not what the fans come to see. That's not part of my personality. I don't like being caught in situations like that," he said. "I apologize to our fans for that kind of show."
Ortiz and his teammates said Saturday afternoon that the incident is a thing of the past and no one expects it to carry over.
"Is it over? Oh it's over. It's over. It's over," Ortiz said. "It's a situation that happened in the game, and whatever happens on the field stays there.
"It's in the past, man. It's behind us. I'm the kind of person that likes to turn the page. I don't like to just keep on throwing wood to the fire. That's not how I play the game."
During Gregg's explanation of the incident on Friday, he questioned Ortiz's baseball ethics, saying he didn't appreciate Ortiz's reaction to his popup. Ortiz fired back on Saturday.
"When was the last time he swung a bat? He don't know nothing about that. When you start situations, you gotta be aware of what's coming up next," Ortiz said.
Gregg also called the Red Sox "whiners" and said "they think they're better than everybody else" because of the club's "$180 million payroll" on Friday.
"If you're getting your a-- kicked, nothing you can do but play better," Ortiz said. "You can be acting stupid out there just because you're getting beat up. That's the reason why you're getting beat up, you're not playing the game the way it's supposed to be (played).
"Play the game the way it's supposed to be and you're not gonna get beat up. But don't be blaming (us and say) 'You guys think you are better than us,' and this and that. I have a lot of friends on the (Orioles), and I don't like to be going through situations like that.
"It's not like we're trying to show up anybody. We got our [butts] kicked the beginning of the season. We don't blame nobody but us."
Ortiz addressed the situation for 10 minutes before getting ready for batting practice on Saturday.
"I apologize to everyone," he said. "Even Gregg. I don't really know him, but he doesn't really seem like he's any crazy person out there."
Joe McDonald and Darren Hartwell cover the Red Sox for ESPNBoston.com.