Word-wise Fully Homomorphic Encryption (FHE) schemes, such as CKKS, are gaining significant traction due to their ability to provide post-quantum-resistant, privacy-preserving approximate computing; an especially desirable feature in Machine-Learning-as-a-Service (MLaaS) cloud-computing paradigms. OpenFHE is a leading CPU-based FHE library with robust CKKS operations, but its server-side performance is not yet sufficient for practical cloud deployment. As GPU computing becomes more common in data centers, many FHE libraries are adding GPU support. However, integrating an efficient GPU backend into OpenFHE is challenging. While OpenFHE uses a Hardware Abstraction Layer (HAL), its flexible architecture sacrifices performance due to the abstraction layers required for multi-scheme and multi-backend compatibility. In this work, we introduce FIDESlib, the first open-source server-side CKKS GPU library that is fully interoperable with well-established client-side OpenFHE operations. Unlike other existing open-source GPU libraries, FIDESlib provides the first implementation featuring heavily optimized GPU kernels for all CKKS primitives, including bootstrapping. Our library also integrates robust benchmarking and testing, ensuring it remains adaptable to further optimization. Furthermore, its software architecture is designed to support extensions to a multi-GPU backend for enhanced acceleration. Our experiments across various GPU systems and the leading open-source CKKS library to date, Phantom, show that FIDESlib offers superior performance and scalability. For bootstrapping, FIDESlib achieves no less than 70x speedup over the AVX-optimized OpenFHE implementation.