The 55-year-old actor, whose wife Hilaria Thomas, is heavily pregnant with their first daughter together, insists they never tweeted from the funeral, telling Vanity Fair magazine: "Never. No. I went to Jimmy Gandolfini's funeral, and when I was there I realised Jimmy Gandolfini didn't have Twitter.
"Jimmy Gandolfini was so beloved as a person, and he was so admired as an actor, and he didn't give a f**k about social media.
"I really learned a lesson at the funeral. I said to myself, 'This is all a waste of time.' Meaning it's fun sometimes, but less and less, and less. It's just another chink in your armour for people to come and kill you. I stopped and said to myself, I'm going to try where I just don't do this anymore."
Alec drew criticism after he tweeted a rant about the journalist, who accused him of sending messages from the funeral, saying: "If put my foot up your f****** a** . . . but I'm sure you'd dig it too much.
"I'm gonna find you. You toxic little queen, and I'm gonna f***...you...up."
The Emmy Award winner, who also has a daughter Ireland, 17, from his first marriage to Kim Basinger, added: "I'm having a baby. And everyone has seen how certain things have played out with my daughter [Ireland], which has been very painful - it's been really unpleasant.
"That has consequences, and I do not want that to happen with my next child. I have one dream in my life and that is that this daughter I'm having - she comes to me about seven or eight years from now, she has a friend, and she's at her house and she says, 'Daddy, Susie's mom says you used to be on TV. Daddy, is that true?' She has no knowledge of me as a public person. That would be heaven for me.
"I'd love to [quit acting] if I could, yeah. That would be the greatest thing in the world."