Michael's definition of legitimacy was contained in this dialog from GF when he went to New Hampshire to woo Kay:
MICHAEL
I'm working for my father now, Kay. He's been sick -- very sick.
KAY
But you're not like him, Michael. I thought you weren't going to become a man like your
father. That's what you told me...
MICHAEL
My father's no different than any other powerful man --
(then, after Kay laughs)
-- Any man who's responsible for other people. Like a senator or a president.
KAY
You know how naive you sound?
MICHAEL
Why?
KAY
Senators and presidents don't have men killed...
MICHAEL
Oh -- who's being naïve, Kay?

In other words, if senators and presidents can have people killed and be considered "legitimate," so can Michael and Vito be considered "legitimate" when they have people killed. No different from other people with responsibilities for other people...