His eyes and look show determination and danger. It means: I'm here. I'm ready to fight. I won't run in fear. Only over my dead body.
Michael Corleone is a fiercely independant, cold-blooded person. He's always deeply thinking about people and situations. He shows extreme determination (close to stubborness sometimes, ie. ex-wife talking about business or visiting kids, but not hot-headed like his brother Santino) and has an very threatening attitude. He has a line of duty/conduct dictated by his values. What he says goes. For him, and for the others too, of course. He's ready to kill/die for what he wants/decides.
"Michael was not tall or heavily built but his presence seemed to radiate danger."
The Godfather by Mario Puzo. (1972, Book Club Associates by arrangement with William Heinemann Ltd), p.135
Michael Corleone has been a highly decorated Marine for almost 5 years. He was a captain, fought in the Pacific and was awarded the Silver Star and the Navy Cross. This excerpt from Military shows how independant he wants to be, with no one deciding for him. No deferment. For Michael Corleone, his country is his family too. You have to protect it.
From this, you can expect him to have been in danger, to have face death, to have killed people. When he realizes that his hands are not shaking, he surely recalls that he's accustomed to danger/threats. He knows he can still be a very dangerous man, even though war is over and he stopped fighting in the Marines.
At the time he visits his father in the hospital, he faces danger, and reacts like an Alpha Male and a Marine. You don't run, you face and you fight. Determination to defend himself and his family, danger and fear have to be for the others. Still calculating his best chances to succeed.