While A Christmas Carol might seem Gothic at first, with its abundance of ghosts and clanking chains, Scrooge doesn't seem to be a Gothic protagonist, not in the traditional sense. Protagonists of Gothic novels tend to be avatars of innocence, and as such are very often young and female. Examples are Jane Eyre, Emily St. Aubert of Ann Radcliffe's The Mysteries of Udolpho, and Laura of Carmilla. These hapless young women are usually plagued and victimized by powerful evils out of their control, which in their turn are almost always metaphors for oppression, hence subgenre designations such as Female Gothic (Jane Eyre, Wuthering Heights) and Irish Gothic (Carmilla). If these protagonists get happy endings, which they don't always, it's by the power of their innocence and goodness.
I say all this because when A Christmas Carol is viewed in this light, Scrooge begins to look a lot like a Gothic villain, representing the evil of avarice. He's the consummate miser at the beginning of the story, and just letting Bob Cratchit stay home on Christmas Day is like pulling teeth for him. Furthermore, the Ghost of Christmas Present implies that Scrooge's unmitigated greed will be the ultimate cause of Tiny Tim's death.
Which brings me to a related point: if Scrooge is the Gothic villain in this story, the evil powerful oppressor, then Tiny Tim and Bob Cratchit represent two different facets of the Gothic protagonist. Tiny Tim is obviously the avatar of innocence, like I mentioned earlier, and Bob Cratchit represents the virtuous, working hero who becomes more common in later Gothic fiction (e.g. Robert Audley of Lady Audley's Secret, Gabriel Utterson of Jekyll and Hyde).
Back to Scrooge, though. To provide a short answer to your question, Scrooge absolutely starts out as a Gothic villain, and he's redeemed in no small part due to the innocence and purity of Tiny Tim and Bob Cratchit.
I hope this helps!