Proof from the book that Jack Ryan learns to lead, do what is necessary and use whatever power is at his disposal.

The book contains several examples of Ryan leading, taking charge of a situation, and using whatever force is necessary to solve the problem.  I believe the best example of this occurs fairly late in the book.  Ryan is already aboard the Red October, and the crew has gotten off of the submarine.  Ryan, Ramius, and a few others are left aboard to bring the secret submarine into the United States.  Ryan hears a gunshot from somewhere in the submarine.  Ryan announces that he is going to go check it out, and Ramius orders him to stay put. 

Ryan stood up. "That may be, Captain, but I know a shot when I hear it." He unbuttoned his jacket and pulled out the pistol.
[...]
"I'm going forward to check."
"You will stay at your post!" Ramius ordered. "You will do as I say!"
"Captain, I just heard something that sounded like gunshots, and I am going forward to check it out. Have you ever been shot at? I have. I have the scars on my shoulder to prove it. You'd better take the wheel, sir."

As the text continues, readers get to see that Ryan is not thrilled about hunting down an armed combatant on an unfamiliar submarine loaded with nuclear weapons. 

"Why me?" he whispered to himself. He'd have to get past thirteen missile tubes to get to the source of that light, cross over two hundred feet of open deck.

Despite his natural inclination to avoid a violent conflict, Ryan leads Ramius toward the threat.  The threat is a GRU intelligence officer named Igor Loginov.  He has been masquerading as Red October's cook, and he is attempting to destroy the vessel.  With no choice but to use deadly force, Ryan shoots and kills Loginov.  

It occurred to Ryan to pray briefly—but for what? For help in killing another man?
[...]
The man was turning as Ryan jerked off six shots. Ryan didn't hear himself screaming. Two rounds connected. The agent was lifted off the deck and twisted halfway around from the impact.