Delivery delay is an important performance metric in opportunistic networks. With given buffer size and copy numbers, how to select appropriate nodes to replicate message is the key to minimizing delivery delay. To solve this problem, this paper proposes an optimal stopping decision method for routing opportunistic networks (OSDR). With OSDR, the average meeting time between a node and the destination is regarded as the forwarding utility of the node. A node carrying a message observes the random forwarding utilities of the nodes it meets, and replicates messages according to the optimal stopping rule, which turns out to be threshold-based. By making tradeoffs between the forwarding utility and waiting cost, OSDR achieves the minimum delivery delay expectation. This paper introduces the OSDR network model and existence proof and calculation of optimal stopping rule in detail. Simulation results show that OSDR outperforms other protocols in delivery delay and delivery rate.