Routing plays a pivotal role in quantum communication as it directly impacts the efficiency, reliability, and scalability of quantum networks. While several studies in the literature have explored routing algorithms leveraging quantum memories, current quantum memory technologies are unable to simultaneously achieve high fidelity, extended storage durations, wide bandwidths, multimode capacity, and high efficiency. To address this limitation, our study focuses on fidelity-guaranteed entanglement routing within a memoryless network architecture, employing both distributed and centralized routing approaches. Utilizing our proposed routing algorithm, MEFID, we achieved a throughput of 72 qubits per second under a fidelity threshold of 0.8 and within three iterative rounds. By integrating a purification process to ensure that the final fidelity consistently exceeds the specified threshold, our algorithm facilitates the development of robust and high-performance quantum networks.