Stellar multiplicity properties have been studied for much of the range from the lowest to the highest stellar masses, but intermediate-mass stars from F-type to late A-type have received relatively little attention. Here we report on a Gemini/NICI snapshot imaging survey of 138 such stars in the young Scorpius-Centaurus (Sco-Cen) region, for the purpose of studying multiplicity with sensitivity down to planetary masses at wide separations. In addition to two brown dwarfs and a companion straddling the hydrogen burning limit we reported previously, here we present 26 new stellar companions and determine a multiplicity fraction within 0.1"--5.0" of 21+/-4%. Depending on the adopted semi-major axis distribution, our results imply a total multiplicity in the range of ~60--80%, which further supports the known trend of a smoothly continuous increase in the multiplicity fraction as a function of primary stellar mass. A surprising feature in the sample is a distinct lack of nearly equal-mass binaries, for which we discuss possible reasons. The survey yielded no additional companions below or near the deuterium-burning limit, implying that their frequency at >200 AU separations is not quite as high as might be inferred from previous detections of such objects within the Sco-Cen region. ; Comment: 23 pages, 5 figures, accepted for publication in ApJ