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Silicon photonics is emerging as a competitive platform for electronic–photonic integrated circuits (EPICs) in the 2 µm wavelength band where GeSn photodetectors (PDs) have proven to be efficient PDs. In this paper, we present a comprehensive theoretical study of GeSn vertical <i>p–i–n</i> homojunction waveguide photodetectors (WGPDs) that have a strain-free and defect-free GeSn active layer for 2 µm Si-based EPICs. The use of a narrow-gap GeSn alloy as the active layer can fully cover entire the 2 µm wavelength band. The waveguide structure allows for decoupling the photon-absorbing path and the carrier collection path, thereby allowing for the simultaneous achievement of high-responsivity and high-bandwidth (BW) operation at the 2 µm wavelength band. We present the theoretical models to calculate the carrier saturation velocities, optical absorption coefficient, responsivity, 3-dB bandwidth, zero-bias resistance, and detectivity, and optimize this device structure to achieve highest performance at the 2 µm wavelength band. The results indicate that the performance of the GeSn WGPD has a strong dependence on the Sn composition and geometric parameters. The optimally designed GeSn WGPD with a 10% Sn concentration can give responsivity of 1.55 A/W, detectivity of 6.12 × 10¹⁰ cmHz^½W⁻¹ at 2 µm wavelength, and ~97 GHz BW. Therefore, this optimally designed GeSn WGPD is a potential candidate for silicon photonic EPICs offering high-speed optical communications.