"Let me say here and now what I should have said on Wednesday," the coach tweeted.

"I sincerely apologize to Courtney Smith and her children for what they have gone through."

As we previously reported ... Meyer was hit with a three-game ban after university officials say he "failed to take sufficient management action" when it came to the handling of assistant coach Zach Smith and his alleged domestic violence toward Courtney.

Meyer caught serious heat when discussing the suspension with reporters ... mostly for the crass tone he took when he was asked if he had a message for Courtney.

Now ... Meyer is realizing he messed up, saying he should have handled that question -- and the entire Zach Smith situation -- differently.

"My words and demeanor on Wednesday did not show how seriously I take relationship violence. I sincerely apologize. I was taught at a very young age that if I ever hit a woman, I would be kicked out of the house and never welcomed back."

"I have the same rule in my house and in the Football Program at Ohio State. Over the years, we have worked hard to educate and remind our coaches and players of the seriousness of relationship violence."

Meyer added, "I understand my lack of more action in this situation has raised concerns about this commitment. I once again apologize for this, and I extend my empathy to all women, men and families who are affected by relationship violence."