-겠지요 means "would" with the nuiance of "right" at the end of English sentences.
As Korean is an agglutinative language, you need to understand each of suffixes or ends in one phrase.
-겠-: similar to "would" in English.
ex. 그거 예쁘겠다. (That would be pretty.)
-지: used to show the speaker thinks something is true.
ex. 이거 좋지. (It's good. (and I think other people would think so.))
-요: a suffix for casual and plain polite speech.
ex. 저 지금 공부해요. (I'm studying now.)
Thus, -겠지요 means all of three above.
이 책은 제 거겠지요. (This book would belong to me. I think it's right.)
이 책은 제 거겠지요? (I think this book belongs to me, right?)
음식을 여기 두면 상하겠지요. (It would go bad if I put it here.)
음식을 여기 두면 상하겠지요? (It would go bad if I put it here, right?)
Hope it helps. :)