Why was Patrick not arrested for his murder?

It was basically not seen as murder by anyone, especially not the police. The way the entire thing played out was that Patrick entered the house and shot various people in self-defense, all while investigating the abduction (and ultimately murder) of a little boy. That's what everyone sees.

Of course we know it basically was cold-blooded murder from Pattrick's side, as the child molester was defenseless, it was an execution. But noone else does. The only one who also knows is Remy Bressant, but he actually believes that Patrick did the right thing. It would be absolutely no problem for the two of them to stage it as self-defense somehow, by claiming the child molester had a weapon (maybe a knife they took from the kitchen) and tried to attack Patrick or something similar, especially since Remy is a cop and people "believe in the police".

But in fact this ties back into a central theme of the entire film. Even if a more thorough investigation might have shown holes in Patrick's and Remy's story, noone actually cared that much. To everyone it was a dead child molester who killed a little boy against a brave hero who saved a cop's ass. The only one who really cares about what he did and if it was the right thing to do is Patrick himself. And you can see that he is slightly distraught when both Angie and other cops congratulate him for what he did. He himself isn't so sure it's worth congratulating.

And this ties right back to the ending of the film, where Patrick is the only one who does not think that the end justifies the means. And his experience with what happened in the house back then and that he might still not be over what he did might actually have contributed to his decision of following the law at the end, even against everyone else's pledges to just leave things be.

Here is a link to reddit which has a thread on this and a good explanaition molester murder. This is a sample:

It's covered much better in the book, so it's more just a glossed over aspect of the movie. Basically in the book, Kenzie and Broussard set it up to cover that aspect as the molestor attacked him and he had no choice but to shoot him. They try to hold him for the murder, but he eventually walks as there is insufficient evidence and a fellow officers word that he didn't execute him.

IMDB also has a good explanation Gone Baby Gone faq.

I think Ghost-E over at reddit has the most plausible answer myself.