Whether Russo would have killed himself, most people relapse. Many recover. Russo might have. He had his girlfriend who had not given up on him.
Russo's girlfriend DID give up on him. She left him and quit her job to go work for the Speaker. She wouldn't even take his phone calls. It was not until Frank stepped up and got Russo to sober up that he was able to get her to talk to him again, and even then she was cautious. Russo was on a dark path all on his own, and Frank redeemed him.
When Russo started to question Frank's motives, Frank would not give him a second chance. The setup with Rachel was entirely in Russo's hands; he could have turned her down and done the right thing and left. Instead, he gave in to his temptations and ruined his life. Russo was causing himself and others pain at this point. When Frank kills him, it is just like the dog in the pilot. He put Russo out of his misery.
1. I'm talking about later in the show, after his girlfriend comes back, she still cares about him when he screws up and goes missing.
Killing a human being who might or might not recover from alcoholism is the same as putting a mortally wounded dog out of its misery? Man, I'm staying out of your way!