The Aprilia rider was fined and hit with a grid place penalty after angrily putting his hand on Morbidelli’s helmet during Saturday practice in Qatar.
Morbidelli later questioned “I wonder what he will say to his kids”, which Espargaro claimed was crossing “a sacred line”.
But tensions have calmed ahead of this weekend’s Valencia MotoGP.
"I went to Morbidelli's motorhome to talk to him and apologise,” Espargaro told Motorsport.
“It's not important if it was my fault or his: I reacted badly.
“Those three seconds project me as a bad person, and I'm not a bad person.
“That's why I sat with him on the sofa and I asked for forgiveness.
“I will work to see how I can better deal with high-tension situations, so that that situation is not repeated.
“The seasons are getting longer, but I have to learn.
“I already work with a psychologist, but it's not like pressing a button.
“I've already changed several things about my environment in the last two months, and that hasn't helped either. Joan (Lacueva), who is a friend of mine and who travelled with me for the last ten years, no longer works with me.
“I hope that in 2024 I will do better.
“I accept the criticism. I know what I will receive for what I will say, but I don't give a shit [about] what they say to me on social media.
"I went to talk to Franco because that's how I felt in the depths of my heart.”
Espargaro arrives for the season-finale feeling the effects of the leg injury sustained last week, which forced an early retirement from the Qatar MotoGP.
He intends to compete in Valencia but it remains to be seen how far his injury will allow him to go.