"A lot of people have been confused or hurt or just do not understand what I was trying to communicate," Bevin said in a video posted on his Twitter page.
"For those of you who have been injured, it is my absolute and sincere apology," said Bevin. "It is not my intention to harm anyone in this process, but to help us all move forward together, we need each other, we are in this together, we are Kentucky."
Expressing his concern for school children who were unable to attend school on Friday due to the strike, Bevin said: "I guarantee you somewhere in Kentucky that a child was sexually assaulted and that they left him at home because there was no nobody there to watch over them. "
" I guarantee you that somewhere today, a child was physically damaged or ingested poison because they were alone at home because a single father had no money to care for them, "he said, adding that "some were introduced to drugs for the first time because they were vulnerable and alone".
His comments faced harsh criticism from Democrats and Republicans, with Republican state Senator Max Wise, who serves as chairman of the chamber's Education Committee, calling them "disgusting" and "reprehensible."
In his apology video, Bevin thanked the people who "understood what I'm saying."
But, he said, "the responsibility of communicating things lies with the person, to a large degree, who is talking, sometimes" When I do, I do it effectively. Sometimes, not so much, and I think this case is an example of the latter. "
CNN's Veronica Stracqualursi contributed to this report.