One primary goal in nanobiotechnology is designing new methodologies for molecular biomedical diagnosis at stages much earlier than currently possible and without use of expensive reagents and sophisticated equipment. In this work, we show the proof of principle for single-molecule detection of the nucleocapsid protein 7 (NCp7), a protein biomarker of the HIV-1 virus, using synthetic nanopores and the resistive-pulse technique. The biosensing mechanism relied upon specific interactions between NCp7 and aptamers of stem-loop 3 (SL3) in the packaging domain of the retroviral RNA genome. One critical step of this study was the choice of the optimal size of the nanopores for accurate, label-free determinations of the dissociation constant of the NCp7 protein - SL3 RNA aptamer complex. Therefore, we systematically investigated the NCp7 protein - SL3 RNA aptamer complex employing two categories of nanopores in a silicon nitride membrane: (i) small, whose internal diameter was smaller than 6 nm, and (ii) large, whose internal diameter was in the range of 7 through 15 nm. Here, we demonstrate that only the use of nanopores with an internal diameter that is smaller than or comparable with the largest cross-sectional size of the NCp7-SL3 aptamer complex enables accurate measurement of the dissociation constant between the two interacting partners. Notably, this determination can be accomplished without the need for prior nanopore functionalization. Moreover, using small solid-state nanopores, we demonstrate the ability to detect drug candidates that inhibit the binding interactions between NCp7 and SL3 RNA by using a test case of N-ethylmaleimide.