Remote entanglement distribution plays a crucial role in quantum networks, which can support many essential and exciting quantum applications. As the network scale expands, it is urgent to design a general and efficient remote entanglement distribution protocol. Nowadays, connection-oriented remote entanglement distribution protocols are available to achieve reliable entanglement distribution. However, in memory-limited quantum networks, connection-oriented entanglement distribution protocols cannot utilize entanglement resources fully and increase the delay of End-to-End (E2E) entanglement connection establishment. To overcome these shortcomings of existing entanglement distribution protocols, we design a connectionless remote entanglement distribution protocol to let Source-Destination (S-D) pairs compete for entanglement resources simultaneously. In our protocol, a fair request scheduling algorithm is proposed to reduce the waiting time without entanglement connections between S-D pairs. Furthermore, a fast scheduling trigger mechanism is proposed to perform entanglement swapping timely to reduce the delay of E2E entanglement connection establishment. The simulation results show that the designed protocol has advantages in terms of resource utilization, throughput, the service completion time of S-D pairs, and the maximum waiting time, compared with the existing remote entanglement distribution protocol.