The efficiency of devices for biomedical applications, including tissue engineering and neuronal stimulation, heavily depends on their biocompatibility and performance level. Therefore, it is important to find adequate materials that meet the necessary requirements such as (i) being intrinsically compatible with biological systems, (ii) providing a sufficient electronic conductivity that promotes efficient signal transduction, (iii) having “soft” mechanical properties comparable to biological structures, and (iv) being degradable in physiological solution. We have developed organic conducting biocompatible single-walled carbon nanotubes (SWCNT) composites based on bovine serum albumin, carboxymethylcellulose, and acrylic polymer and investigated their properties, which are relevant for biomedical applications. This includes ζ-potential measurements, conductivity analyses, and SEM micrographs, the latter providing a local analysis of SWCNT distribution in the base material. We observed the development of the electrical conductivity of the SWCNT composites exposed to 1 mM KCl electrolyte for 40 days, representing a high stability of the samples. The conductivity of samples reaches 1300 S/m for 0.45 wt.% nanotubes. Moreover, we demonstrated the biocompatibility of the composites via cultivating fibroblast cell culture. Finally, we showed that composite coating results in the longer lifespan of cells on the surface. Overall, the SWCNT-based conductive composites might be a promising material for extended biomedical applications.