According to multiple media reports, Kahne posted messages on his account that he saw a mother breastfeeding a child in a supermarket, calling it "nasty" and saying he didn't "feel like shopping any more or eating."
Reports say Kahne then sent a crude reply message to a Twitter user who took issue with his comments.
After apparently deleting the offending posts from his Twitter feed, Kahne posted an apology on his Facebook page Wednesday, saying it wasn't his intention "to offend any mother who chooses to breastfeed her child, or, for that matter, anyone who supports breast feeding children. I want to make that clear."
Kahne said he understands his comments were "offensive to some people" and apologized.
"In all honestly, I was surprised by what I saw in a grocery store," Kahne said. "I shared that reaction with my fans on Twitter. It obviously wasn't the correct approach, and, after reading your feedback, I now have a better understanding of why my posts upset some of you.
"My comments were not directed at the mother's right to breastfeed. They were just a reaction to the location of that choice, and the fashion in which it was executed on that occasion."
Kahne said he respects the mother's right to feed her child "whenever and wherever she pleases."
Kahne also posted a personal reply to the user he'd insulted.
"I wanted to apologize for saying what I said to you yesterday," Kahne wrote. "It was out of line."
Kahne's team said its "concerns have been conveyed" to Kahne.
"We appreciate that he chose to follow up with his fans and others who were upset by the comments," Hendrick Motorsports said in a statement.
NASCAR did not immediately respond to a request for comment.