Braden and Rodriguez clashed in an April 22 game at the Oakland Coliseum when A-Rod passed over the mound while returning to first base on a foul ball, which Braden perceived as a violation of the game's unwritten code.
On Wednesday, Braden fired another notable remark, when asked by a reporter from the ComCast Bay Area network if he and Rodriguez would have to "settle [things] with knuckles."
"Obviously there's things that are going to have to happen -- just out of respect to my teammates, out of respect to the game," Braden replied.
Rodriguez was apprised of Braden's comments by Yankees director of media relations Jason Zillo, and with a smile, A-Rod said that he found the words as amusing as he did weeks ago.
"Look, it's tempting to sit here and go back and forth in the media for the next three months, but I'm not going to do that," Rodriguez said. "For us to be thinking about him and Oakland would be a disservice for us. We have the Red Sox and a big series here."
The Yankees meet the Athletics from July 5-7 in Oakland, and Braden has made it no secret that he won't be asking A-Rod to sign a few baseballs during that trip.
"I'm not a fan of his antics," Braden told the television network. "Obviously what he displayed wasn't the first display of his lack of respect for the game or those playing it, so that never sat well with me. He's just an individualistic player.
"That's his approach to the game, he plays for the name on the back, and I don't know if he's noticed, but he doesn't have a name on the back over there. He should play for the name on the front a little more."
A-Rod dismissed Braden in Oakland as a pitcher with only "a handful of wins," but said Friday that he did not want to extend the spat further. He mentioned Major League Baseball discipline czar Bob Watson and said that Watson will "read all this stuff and he'll handle it."
"I think Major League Baseball reads the same articles as we do," Rodriguez said. "I really don't want to extend his extra 15 minutes of fame he already has."
Yankees manager Joe Girardi also said that he would not focus on Braden's comments or intentions.
"We don't play them until July," Girardi said. "I worry about our club and what our guys say. I'm not really concerned with what other guys say. July is a long ways away, so I'm not really worried about it."
Rodriguez said that he is sorry the incident took place in Oakland.
"There's probably some overreaction, but at the end of the day, it was just an unfortunate incident," Rodriguez said. "We're past it."
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.