The data rate of a visible light communication (VLC) system is basically determined by the electrical-to-optical (E-O) bandwidth of its light-emitting diode (LED) source. In order to break through the intrinsic limitation of the carrier recombination rate on E-O bandwidth in conventional c-plane LEDs based on InGaN quantum wells, a blue micro-LED with an active region of nano-structured InGaN wetting layer is designed, fabricated, and packaged to realize a high-speed VLC system. The E-O bandwidth of the micro-LED can reach up to 1.3 GHz. Based on this high-speed micro-LED, we demonstrated a data rate of 2 Gbps with a bit error rate (BER) of $1.2\times {10}^{-3}$ with simple on-off keying signal for a 3-m real-time VLC. In addition, a 4-Gbps VLC system using quadrature phase shift keying-orthogonal frequency-division multiplexing with a BER of $3.2\times {10}^{-3}$ is also achieved for the same scenario. Among all the point-to-point VLC systems based on a single-pixel LED, this work has the highest distance-bandwidth product of 3 GHz·m and the highest distance-rate product of 12 Gbps·m.