Condolences from a student to a professor I have a professor I really care about and our professional relationship as a student/teacher has been growing in a positive direction. The professor has been really kind to me in many ways including giving personal/academic advise and providing motivation towards certain goals. 

My professor has been gone, without word, for an extended period of time...

I got stupid and curious, and using Google, I figured out the professor lost an important loved one. 

I now feel ashamed because I know I violated the professors privacy; however, now that I know my first instinct is of empathy and to give my condolences to my professor. 

Now the more I think about it, I don’t want to say anything so that I don’t disrespect our relationship, but I feel that giving my condolences is the “right” thing to do when someone you care about is in pain, but I only am aware of the situation because I already disrespected our relationship. 

I’m frustrated and confused. Could any lecturers or professors give me some guidance?
I'm not sure that you 'violated the professor's privacy'. Googling his name seems altogether appropriate.

Still, I wouldn't say anything. Since he hasn't told you (and no one else has told you), it seems he doesn't want to discuss it with you. I would respect this. Thinking back to when I taught classes, I certainly would have appreciated the effort, but it also would make me just a tiny amount uncomfortable, particularly since I hadn't announced the death. 

By the way, I'm assuming you're an undergraduate and you only know him as a teacher. If you are a grad student and he is a prof in your subfield, I still would err on the side of not saying anything, but it could go either way depending on the personalities involved.