I'm just looking for canon, either books or JKR. Or movies, I haven't seen those.
In a July 16, 2005 interview, JKR was asked
MA: ... “Do the memories stored in a Pensieve reflect reality or the views of the person they belong to?”
JKR: It’s reality. ...
Now, when the incident happened where Sirius was imprisoned for killing Peter & muggles, he knew who the real secret-keeper was and saw what happened when confronting Peter about it.
Both Sirius and Peter were members of the Order of the Phoenix, of which Dumbledore was the founder. He was also already Headmaster at Hogwarts and Order of Merlin, First Class. He also might have been the Supreme Mugwump and Chief Warlock (but I can't find canon for that). Either way, he had pull.
Fudge was only a Junior Minister and didn't actually witness what happened between Sirius and Peter, only the aftermath, so I can't imagine he had more pull than Dumbledore.
After his death, his sentence was revoked, so it's possible for sentences to be overturned. Dumbledore thought Sirius was guilty, and we know that veritaserum wouldn't be useful. But, since we know the Pensieve exists and (if at Hogwarts) Dumbledore would have access to it, no matter what Dumbledore believes, he had access to obtain the truth about what happened between two of his members of the Order.
Also, there is a possibility of using memories as evidence.
So, is there any canon as to why Dumbledore didn't go to Azkaban to find the truth out about what happened between Sirius and Peter with the Pensieve (assuming Black would cooperate to exonerate himself)?