Believe it or not, guys can be shy. The next time you see him smile and say hello. Introduce yourself and start a conversation. See where it goes from there.
Go to your personnel manager and explain your case; ask that he be spoken to discreetly. Staring is a form of sexual harassment and should not be tolerated.
If you aren't in a relationship and he has no ring, I would try to talk to him. I would say maybe start out by walking by and just saying hi when he looks at you.
