Pretending that this is 10/20 cents, not dollars, and thus my opinion has some merit, my dominant thought is what is the probability that he will call me with a weaker hand and is this guy cagey enough to play THAT slow with a big hand. I see three reasonable scenarios:

1) he tried to buy the pot and has given up
2) he had a big wrap/flush draw
3) he has quad Ks (maybe, under boat but i doubt it)

If it's (1) or (2) he probably won't call and if it's (3) you're dead. Your only real hope for more money is that he caught a backdoor Q/J flush. I don't think this is significantly more likely than a slow played boat/quads or a re-raise bluff. I check.

Of course, I realize that river play on the 10/20 dollar tables is completely different from what I'm used to.