Cross-domain recommendation (CDR) is a task that aims to improve the recommendation performance in a target domain by leveraging the information from source domains. Contrastive learning methods have been widely adopted among intra-domain (intra-CL) and inter-domain (inter-CL) users/items for their representation learning and knowledge transfer during the matching stage of CDR. However, we observe that directly employing contrastive learning on mixed-up intra-CL and inter-CL tasks ignores the difficulty of learning from inter-domain over learning from intra-domain, and thus could cause severe training instability. Therefore, this instability deteriorates the representation learning process and hurts the quality of generated embeddings. To this end, we propose a novel framework named SCCDR built up on a separated intra-CL and inter-CL paradigm and a stop-gradient operation to handle the drawback. Specifically, SCCDR comprises two specialized curriculum stages: intra-inter separation and inter-domain curriculum scheduling. The former stage explicitly uses two distinct contrastive views for the intra-CL task in the source and target domains, respectively. Meanwhile, the latter stage deliberately tackles the inter-CL tasks with a curriculum scheduling strategy that derives effective curricula by accounting for the difficulty of negative samples anchored by overlapping users. Empirical experiments on various open-source datasets and an offline proprietary industrial dataset extracted from a real-world recommender system, and an online A/B test verify that SCCDR achieves state-of-the-art performance over multiple baselines.