Graphene has been synthesized on 100- and 300-nm-thick Ni/$SiO_2$/Si substrates with $CH_4$ gas (1 SCCM) diluted in mixed gases of 10% $H_2$ and 90% Ar (99 SCCM) at $900^{circ}C$ by using inductively-coupled plasma chemical vapor deposition (ICP-CVD). The film morphology of 100-nm-thick Ni changed to islands on $SiO_2$/Si substrate after heat treatment at $900^{circ}C$ for 2 min because of grain growth, whereas 300-nm-thick Ni still maintained a film morphology. Interestingly, suspended graphene was formed among Ni islands on 100-nm-thick Ni/$SiO_2$/Si substrate for the very short growth of 1 sec. In addition, the size of the graphene domains was much larger than that of Ni grains of 300-nm-thick Ni/$SiO_2$/Si substrate. These results suggest that graphene growth is strongly governed by the direct formation of graphene on the Ni surface due to reactive carbon radicals highly activated by ICP, rather than to well-known carbon precipitation from carbon-containing Ni. The D peak intensity of the Raman spectrum of graphene on 300-nm-thick Ni/$SiO_2$/Si was negligible, suggesting that high-quality graphene was formed. The 2D to G peak intensity ratio and the full-width at half maximum of the 2D peak were approximately 2.6 and $47cm^{-1}$, respectively. The several-layer graphene showed a low sheet resistance value of $718{Omega}/sq$ and a high light transmittance of 87% at 550 nm.