In Internet of Things (IoT) applications, system-on-chip (SoCs) with embedded processors are widely used. As an embedded processor, RISC-V, which is license-free and has an extensible instruction set, is receiving attention. However, designing such embedded processors requires an enormous effort to achieve a highly efficient microarchitecture in terms of performance, power consumption, and circuit area, as well as the design verification of running complex software, including modern operating systems such as Linux. In this paper, we propose a method for directly describing the RTL structure of a pipelined RISC-V processor with cache memories, a memory management unit (MMU), and an AXI bus interface using the C++ language. This pipelined processor C++ model serves as a functional simulator of the complete RISC-V core, whereas our C2RTL framework translates the processor C++ model into a cycle-accurate RTL description in the Verilog-HDL and RTL-equivalent C model. Our processor design methodology using the C2RTL framework is unique compared to other existing methodologies because both the simulation and RTL models are derived from the same C++ source, which greatly simplifies the design verification and optimization processes. The effectiveness of our design methodology is demonstrated on a RISC-V processor that runs Linux OS on an FPGA board, achieving a significantly short simulation time of the original C++ processor model and RTL-equivalent C model in comparison to a commercial RTL simulator.