a word for the act of looking at one's navel in contemplation

omphalaskepsis is the word you are looking for. It comes from Greek omphalos (navel), and skepsis (examination).

Apparently, the practice of omphalaskepsis is supposed to help meditation and give you a deeper understanding of life, divinity and what-not. The first time I heard this was in a yoga-like meditation class, but it still seems an odd thing to me, since in Spain someone who sits around all day "looking at his/her bellybutton" is a world-class lazybones...