The emergence of the SARS-CoV-2 virus and the associated pandemic has affected the entire human population. Human susceptibility to the virus has highlighted a tremendous need for affordable diagnostic systems to manage the pandemic and monitor the effectiveness of vaccination. We have developed a simple and label-free electrochemical immunosensor for the detection of human anti-SARS-CoV-2 IgG antibodies, which consists of a supporting screen-printed carbon electrode (SPCE) modified with an electrodeposited polyaniline film and glutaraldehyde, allowing effective immobilization of the SARS-CoV-2 spike glycoprotein receptor-binding domain (RBD) as a biorecognition element. The impedimetric immunosensor showed a linear response over a wide concentration range of 0.01–10 μg mL−1, that is, 67 pM–6.7 nM, with a low detection limit of 25.9 pM. A dual working electrode configuration with a built-in negative control unit was demonstrated for practical field applications. The immunosensor was successfully used in a real serum sample from an infected patient and showed good reproducibility and fair agreement with ELISA. An optional amplification step with secondary goat anti-human IgG antibodies was demonstrated, resulting in an extended linear range and a detection limit as low as 0.93 pM.