We demonstrate the growth of multilayer and single-layer graphene on copper foil using bipolar pulsed direct current (DC) magnetron sputtering of a graphite target in pure argon atmosphere. Single-layer graphene (SG) and few-layer graphene (FLG) films are deposited at temperatures ranging from 700 degrees C to 920 degrees C within <30 min. We find that the deposition and post-deposition annealing temperatures influence the layer thickness and quality of the graphene films formed. The films were characterized using atomic force microscopy (AFM), scanning electron microscopy (SEM), high-resolution transmission electron microscopy (HRTEM), Raman spectroscopy, X-ray photoelectron spectroscopy (XPS), and optical transmission spectroscopy techniques. Based on the above studies, a diffusion-controlled mechanism was proposed for the graphene growth. A single-step whole blood assay was used to investigate the anticoagulant activity of graphene surfaces. Platelet adhesion, activation, and morphological changes on the graphene/glass surfaces, compared to bare glass, were analyzed using fluorescence microscopy and SEM techniques. We have found significant suppression of the platelet adhesion, activation, and aggregation on the graphene-covered surfaces, compared to the bare glass, indicating the anticoagulant activity of the deposited graphene films. Our production technique represents an industrially relevant method for the growth of SG and FLG for various applications including the biomedical field.