Elliptic Curve Cryptography (ECC) is an encryption method that provides security comparable to traditional techniques like Rivest-Shamir-Adleman (RSA) but with lower computational complexity and smaller key sizes, making it a competitive option for applications such as blockchain, secure multi-party computation, and database security. However, the throughput of ECC is still hindered by the significant performance overhead associated with elliptic curve (EC) operations. This paper presents gECC, a versatile framework for ECC optimized for GPU architectures, specifically engineered to achieve high-throughput performance in EC operations. gECC incorporates batch-based execution of EC operations and microarchitecture-level optimization of modular arithmetic. It employs Montgomery's trick to enable batch EC computation and incorporates novel computation parallelization and memory management techniques to maximize the computation parallelism and minimize the access overhead of GPU global memory. Also, we analyze the primary bottleneck in modular multiplication by investigating how the user codes of modular multiplication are compiled into hardware instructions and what these instructions' issuance rates are. We identify that the efficiency of modular multiplication is highly dependent on the number of Integer Multiply-Add (IMAD) instructions. To eliminate this bottleneck, we propose techniques to minimize the number of IMAD instructions by leveraging predicate registers to pass the carry information and using addition and subtraction instructions (IADD3) to replace IMAD instructions. Our results show that, for ECDSA and ECDH, gECC can achieve performance improvements of 5.56x and 4.94x, respectively, compared to the state-of-the-art GPU-based system. In a real-world blockchain application, we can achieve performance improvements of 1.56x, compared to the state-of-the-art CPU-based system.