If you bet the flop you HAVE to bet the turn. You are betting into 3 other players with a hand that has only two outs to improve if you are behind, so you have to either check/fold the flop or decide to carry through your charade at least through the turn. The chance that everyone just folds to your flop bet at these limits is very low, so there is not much point in firing once if you won't fire again.

But since you didn't play it that way, you have to call the river bet. Since the turn got checked around and you weakly checked the river, you don't know whether you just induced a bluff to the late position player who just witnessed two rounds of weakness. Good chance he has one of two hands : nothing or a weak ace. With you getting 4.5:1 odds on a river call, you need to call here. So no, you shouldn't have folded.