I don't think I've heard Diablo called a roguelike before. It's got the dungeon crawling and procedural generation, but the permanent death seems to be one of the most quintessential features of the genre. Diablo does not have that.
If you choose to define Roguelike to include permadeath or that it must be turn based, then it is not, of course. But if Diablo is not a roguelike, its next door neighbor is.
Another comment linked the creator's postmortem; he said he was influenced by roguelikes but specifically called out that Diablo doesn't have permadeath as opposed to roguelikes. His original design doc included permadeath, but it didn't make it far into production.
Coincidentally, I played Diablo II with a self-imposed permanent death rule several years ago. I made it to the... Desert, I think it was? and then got killed and never really got back into the game. It just wasn't designed for that.
I never considered permadeath an important part of Angband or any other roguelike because I always scummed.