Dale apologised to Holmes as well as his family and work colleagues for his "frankly idiotic" behaviour.
He also said he will make a donation to a charity of Holmes' choosing.
The incident happened after Dale apparently tried to move Holmes out of the way during an interview with Damian McBride on ITV's Daybreak.
The LBC Drive presenter, who is the publisher of McBride's explosive memoirs, was captured on film grappling with the campaigner, who was holding anti-nuclear banners alongside his dog.
The barking of the protester's dog, who is also named Stuart, could be heard by TV viewers as the struggle continued.
After a few moments, the pair separated and dusted themselves down, and Holmes went back to trying to edge his way into view of the cameras.
Hours after the event Dale wrote: "I was determined this idiot shouldn’t disrupt what was an important interview for my author.
"Anyone who has seen the pictures and video can see that there was no real violence. I certainly didn’t hurt the guy. He threw a punch at me but missed, and the only injury was when the man’s dog bit him on the bum."
Dale added: "Everyone has an inalienable right to protest, but no one has a right to make a continual nuisance of themselves and interrupt interviews like that."
However he took a different tone in his apology on Thursday:
Following the incident on Brighton seafront on Tuesday morning, I have today voluntarily attended Brighton police station where I accepted a police caution. The police have informed me they now regard the matter as closed. I want to thank them for the fair and courteous way they have dealt with me throughout.
But above all I want to issue this public apology for my behaviour.
I want to apologise and say sorry to Stuart Holmes, who is a passionate campaigner and well known to everyone who attends party conferences and was perfectly entitled to do as he did on Tuesday in trying to get attention for his causes. It was totally out of character for me to react to him in the way I did.
I also want to apologise for the blogpost I wrote after the incident. It was full of absurd bravado and in the heat of the moment I behaved in a frankly idiotic way.
I have embarrassed not only myself but my family and my work colleagues and I apologise to them.
I also want to apologise to Labour leader Ed Miliband and his conference attendees.
I did apologise personally to Mr Holmes on Tuesday afternoon and we shook hands. He agreed to let the matter rest, but I have no complaint that he changed his mind on reflection.
Since the events of Tuesday I have gone through what happened over and over again in my mind. Whatever I felt at the time, nothing can justify what I did.
In addition, having accepted my guilt, I feel I should make some sort of reparation to Mr Holmes. I will pay for a new placard for him and also make a donation to a charity of his choice.
Finally, people have questioned why I didn’t remove the blogpost and why I have said nothing more until now. On the latter point, I was advised not to because the police were involved. On the first point, I felt it important people should be able to have their say. I will have to live with the justified criticisms for a long time.
I know there will be many who will never forgive me for what I did and I understand that, but those who know me will know that I mean every word of my apology to Mr Holmes, Mr Miliband, the Police, my family, friends and colleagues.