No, the character didn't change
Michael's ruthlessness was rather consistent with his father Vito.
In II, the murder of the "Black Hand" guy was in response to a threat of violence -- Vito had seen the man hold a knife to the throat of the young actress so his murder was not just for business reasons.
Although that indeed contributed to Vito's decision to kill Don Fanucci, it wasn't the trigger. The trigger was that Fanucci tried to racketeer the trio Vito/Clemenza/Tessio ("wet his beak"), and Vito decided to kill many birds with one stone:
- Get rid of the racketeering problem (which was worrying himself as well as the whole neighborhood)
- Earn respect of Clemenza/Tessio and become the leader of the band
- Earn respect of the neighborhood, essentially make bones and set to become a Don
- Get rid of the asshole
So, although Fanucci was not a serious life threat to Vito, killing him solved many problems and created many benefits. And the fact that he was a bad person sort of negated any possible pangs of conscience.
Now, Moe Greene wasn't killed purely (or, I would even say, substantially) for business reasons. The trigger was that he had sided with Barzini (was stupid enough to reveal it himself). He was also blatantly disrespectful of the Corleones (slapped Fredo in public, said "you Guineas make me laugh", "you don't have that muscle anymore" etc.).
So, aside of the benefit of acquiring Greene's assets from his heirs for cheap, the benefit of killing him was to show everyone in the crime world that the new Corleone boss was powerful and was to be feared pretty much like the old Don.
It could be argued that Michael could kill just the NY crime bosses and spare Moe Greene giving him another chance to show respect and concede. But Michael was just insulted enough not to, and decided to get rid of him as, otherwise, he could hold a grudge and retaliate later.